1off echo; 2 3 4=============================================================== 5 6-------------------------------------------------------------------------- 7 8This is LIEPDE - a program for calculating infinitesimal symmetries 9of single differential equations or systems of de's 10 11The ODE under investigation is : 12 13 2 2 3 14y = - f *y - 3*y *f - y *y - 2*f *y - f*y + y 15 2x x x x 16 17for the function(s) : 18 19y(x) 20 21 22Start of splitting equations 234 equations result 24 25~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 26This is CRACK - a solver for overdetermined partial differential equations 27 28new function: c_1(x) 29new function: c_2(x) 30Integration of e_1 31 32Substitution of xi_x(x,y) by an expression in 33c_2(x), c_1(x) found in e_1 : 34xi_x= - (c_1 + c_2*y) 35 36 37new function: c_3(x) 38new function: c_4(x) 39Integration of e_2 40 41Substitution of eta_y(x,y) by an expression in 42c_2(x), c_4(x), c_3(x) found in e_2 : 13 factors in 6 terms 43 44Separation of e_3 yields (e_5 e_7 e_6) 45 46Separation of e_4 yields (e_8 e_12 e_9 e_11 e_10) 47 48e_6 is a consequence of e_9. 49Substitution of c_2(x) found in e_5 : 50c_2=0 51 52 53Substitution of c_4(x) by an expression in 54c_1(x) found in e_9 : 55c_4= - c_1 56 x 57 58 59Substitution of c_3(x) by an expression in 60c_1(x) found in e_10 : 61 - c_1 + c_1 *f + f *c_1 62 2x x x 63c_3=---------------------------- 64 3 65 66 67new constant: c_5 68Not enough integrations to solve for a function. 69new constant: c_6 70Integration of e_7 71 72c_1 73 3x 74 75from e_7 and e_11. e_11: 76 77 2 780=2*c_1 + 5*c_1 *f - 4*c_1 *f - 7*c_1 *f - 2*f *c_1 - 7*f *c_1*f 79 3x 2x x x x 2x x 80 81is replaced by e_13: 82 830=c_1 - c_1 *f - f *c_1 84 2x x x 85 86 87new constant: c_7 88Not enough integrations to solve for a function. 89new constant: c_8 90Integration of e_13 91 92c_1 93 x 94 95from e_13 and e_7. e_7: 96 970=c_1 - c_1*f + c_6 98 x 99 100is replaced by e_14: 101 1020=c_6 - c_8 103 104 105c_1 106 4x 107 108from e_13 and e_12. e_12: 109 11 terms, with derivatives of functions of all 1 variables: x 110 111 (c_1 ,c_1 ,c_1 ,c_1 ,c_1) 112 4x 3x 2x x 113 114is replaced by e_15: 115 8 terms, with derivatives of functions of all 1 variables: x 116 117 (c_1 ,c_1 ,c_1 ,c_1) 118 3x 2x x 119 120 121c_1 122 3x 123 124from e_13 and e_15. e_15: 125 8 terms, with derivatives of functions of all 1 variables: x 126 127 (c_1 ,c_1 ,c_1 ,c_1) 128 3x 2x x 129 130is replaced by e_16: 131 1320=c_1 - c_1 *f - f *c_1 133 2x x x 134 135 136new constant: c_9 137Not enough integrations to solve for a function. 138new constant: c_10 139Integration of e_16 140 141c_1 142 x 143 144from e_16 and e_13. e_13: 145 1460=c_1 - c_1*f + c_8 147 x 148 149is replaced by e_17: 150 1510=c_10 - c_8 152 153 154 155>>>>>>>>> Solution : 156This is the end of the CRACK run 157~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 158 159-------- Symmetries: 160 161xi_x=3*c_1 162 163eta_y= - c_1 + c_1 *f - 3*c_1 *y + f *c_1 164 2x x x x 165 166with c_10, c_1(x) 167which still have to satisfy: 168 169 1 1700=3*(c_1 - c_1*f - ---*c_10) 171 x 3 172 173 174-------- 175 176=============================================================== 177 178-------------------------------------------------------------------------- 179 180This is LIEPDE - a program for calculating infinitesimal symmetries 181of single differential equations or systems of de's 182 183The PDE under investigation is : 184 185 2 186u =u + u 187 t 2x x 188 189for the function(s) : 190 191u(t,x) 192 193 194Start of splitting equations 1953 equations result 196 197~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 198This is CRACK - a solver for overdetermined partial differential equations 199 200new function: c_11(t,x,u,u!`2,u!`2!`2) 201new function: c_12(t,x,u,u!`2,u!`2!`2) 202Integration of e_1 203 204Substitution of eta_u(t,x,u,u!`2,u!`2!`2,u!`2!`2!`2) by an expression in 205c_12(t,x,u,u!`2,u!`2!`2), c_11(t,x,u,u!`2,u!`2!`2) found in e_1 : 206eta_u= - (c_11 + c_12*u`2`2`2) 207 208 209Separation of e_2 yields (e_4 e_5) 210 211Separation of e_3 yields (e_6 e_7 e_9 e_8) 212 213new function: c_13(t,x,u,u!`2) 214Integration of e_4 215 216Substitution of c_12(t,x,u,u!`2,u!`2!`2) by an expression in 217c_13(t,x,u,u!`2) found in e_4 : 218c_12= - c_13 219 220 221Separation of e_5 yields (e_10 e_11) 222 223new function: c_14(t,x,u,u!`2) 224new function: c_15(t,x,u,u!`2) 225Integration of e_7 226 227Substitution of c_11(t,x,u,u!`2,u!`2!`2) by an expression in 228c_15(t,x,u,u!`2), c_14(t,x,u,u!`2) found in e_7 : 229c_11= - (c_14 + c_15*u`2`2) 230 231 232Separation of e_8 yields (e_12 e_13 e_14) 233 234Separation of e_9 yields (e_15 e_16 e_18 e_17) 235 236new function: c_16(t,x,u) 237Integration of e_10 238 239Substitution of c_13(t,x,u,u!`2) by an expression in 240c_16(t,x,u) found in e_10 : 241c_13= - c_16 242 243 244Separation of e_11 yields (e_20 e_19) 245 246new function: c_17(t,x,u) 247new function: c_18(t,x,u) 248Integration of e_15 249 250Substitution of c_15(t,x,u,u!`2) by an expression in 251c_18(t,x,u), c_17(t,x,u) found in e_15 : 252c_15= - (c_17 + c_18*u`2) 253 254 255Separation of e_14 yields (e_23 e_21 e_22) 256 257Substitution of c_18(t,x,u) by an expression in 258c_16(t,x,u) found in e_13 : 259c_18=3*c_16 260 261 262new function: c_19(t,u) 263Integration of e_20 264 265Substitution of c_16(t,x,u) by an expression in 266c_19(t,u) found in e_20 : 267c_16= - c_19 268 269 270new function: c_20(t,x) 271Integration of e_22 272 273Substitution of c_17(t,x,u) by an expression in 274c_20(t,x) found in e_22 : 275c_17= - c_20 276 277 278new function: c_21(t) 279Integration of e_19 280 281Substitution of c_19(t,u) by an expression in 282c_21(t) found in e_19 : 283c_19= - c_21 284 285 286new function: c_22(t) 287Integration of e_23 288 289Substitution of c_20(t,x) by an expression in 290c_21(t), c_22(t) found in e_23 : 291 - (c_21 *x + c_22) 292 t 293c_20=--------------------- 294 2 295 296 297new function: c_23(t,x,u) 298new function: c_24(t,x,u) 299Integration of e_16 300 301Substitution of c_14(t,x,u,u!`2) by an expression in 302c_24(t,x,u), c_23(t,x,u), c_21(t), c_22(t) found in e_16 : 14 factors in 6 terms 303 304Separation of e_17 yields (e_24 e_25) 305 306Separation of e_18 yields (e_29 e_26 e_28 e_27) 307 308new function: c_25(t,x) 309Integration of e_24 310 311Substitution of c_24(t,x,u) by an expression in 312c_25(t,x) found in e_24 : 313c_24= - c_25 314 315 316new function: c_26(t) 317Integration of e_25 318 319Substitution of c_25(t,x) by an expression in 320c_21(t), c_22(t), c_26(t) found in e_25 : 321 2 322 - c_21 *x - 2*c_22 *x - 2*c_26 323 2t t 324c_25=----------------------------------- 325 8 326 327 328new function: c_27(t,x) 329new function: c_28(t,x) 330Integration of e_27 331 332Substitution of c_23(t,x,u) by an expression in 333c_28(t,x), c_27(t,x) found in e_27 : 334 u 335 - (e *c_28 + c_27) 336c_23=--------------------- 337 u 338 e 339 340 341Separation of e_29 yields (e_31 e_30) 342 343new function: c_29(t) 344Integration of e_28 345 346Substitution of c_28(t,x) by an expression in 347c_21(t), c_22(t), c_26(t), c_29(t) found in e_28 : 348 3 2 349 - c_21 *x + 6*c_21 *x - 3*c_22 *x - 6*c_26 *x - 3*c_29 350 3t 2t 2t t 351c_28=-------------------------------------------------------------- 352 48 353 354 355Separation of e_30 yields (e_33 e_32 e_35 e_34) 356 357new constant: c_30 358new constant: c_31 359new constant: c_32 360Integration of e_33 361 362Substitution of c_22(t) by an expression in 363c_32, c_31, c_30 found in e_33 : 364 2 365 - 2*c_30 - c_31*t - 2*c_32*t 366c_22=-------------------------------- 367 2 368 369 370new constant: c_33 371new constant: c_34 372new constant: c_35 373new constant: c_36 374Integration of e_32 375 376Substitution of c_21(t) by an expression in 377c_36, c_35, c_34, c_33 found in e_32 : 378 2 3 379 - 6*c_33 - 3*c_34*t - c_35*t - 6*c_36*t 380c_21=-------------------------------------------- 381 6 382 383 384new constant: c_37 385new constant: c_38 386Integration of e_34 387 388Substitution of c_26(t) by an expression in 389c_35, c_38, c_37 found in e_34 : 390 2 391c_26= - (c_35*t + c_37 + c_38*t) 392 393 394new constant: c_39 395Integration of e_35 396 397Substitution of c_29(t) by an expression in 398c_31, c_39 found in e_35 : 399c_29= - 2*c_31*t - c_39 400 401 402 403>>>>>>>>> Solution : 404This is the end of the CRACK run 405~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 406 407-------- 1. Symmetry: 408 409xi_t=0 410 411xi_x=0 412 413eta_u=1 414 415-------- 2. Symmetry: 416 417xi_t=0 418 419xi_x=0 420 421eta_u=u 422 x 423 424-------- 3. Symmetry: 425 426xi_t=0 427 428xi_x=0 429 430eta_u=2*u *t + x 431 x 432 433-------- 4. Symmetry: 434 435xi_t=0 436 437xi_x=0 438 439 3 440eta_u=u + 3*u *u + u 441 3x 2x x x 442 443-------- 5. Symmetry: 444 445xi_t=0 446 447xi_x=0 448 449 2 2 3 2 2 2 450eta_u=4*u *t + 12*u *u *t + 4*u *t*x + 4*u *t + 4*u *t*x + u *x - x 451 3x 2x x 2x x x x 452 453-------- 6. Symmetry: 454 455xi_t=0 456 457xi_x=0 458 459 3 3 2 3 3 2 2 2 460eta_u=8*u *t + 24*u *u *t + 12*u *t *x + 8*u *t + 12*u *t *x + 12*u *t 461 3x 2x x 2x x x x 462 463 2 3 464 + 6*u *t*x + 6*t*x + x 465 x 466 467-------- 7. Symmetry: 468 469xi_t=0 470 471xi_x=0 472 473 3 2 474eta_u=2*u *t + 6*u *u *t + u *x + 2*u *t + u *x 475 3x 2x x 2x x x 476 477-------- 8. Symmetry: 478 479xi_t=0 480 481xi_x=0 482 483 2 484eta_u=u + u 485 2x x 486 487-------- 9. Symmetry: 488 489xi_t=0 490 491xi_x=0 492 493 2 2 2 2 494eta_u=4*u *t + 4*u *t + 4*u *t*x + 2*t + x 495 2x x x 496 497-------- 10. Symmetry: 498 499xi_t=0 500 501xi_x=0 502 503 2 504eta_u=2*u *t + 2*u *t + u *x 505 2x x x 506 507-------- Further symmetry: 508 509xi_t=0 510 511xi_x=0 512 513 c_27 514eta_u=------ 515 u 516 e 517 518with c_27(t,x) 519which still has to satisfy: 520 5210=c_27 - c_27 522 t 2x 523 524 525-------- 526 527=============================================================== 528 529-------------------------------------------------------------------------- 530 531This is LIEPDE - a program for calculating infinitesimal symmetries 532of single differential equations or systems of de's 533 534The PDE under investigation is : 535 536 2 537u =u - u 538 2x t x 539 540for the function(s) : 541 542u(t,x) 543 544##### The function eta_u that is to be computed depends on (u`2`2) 545 which is a left hand side of the input system or a 546derivative of a left hand side of the input system 547Start of splitting equations 5483 equations result 549 550~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 551This is CRACK - a solver for overdetermined partial differential equations 552 553new function: c_40(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2,u!`1!`1!`2) 554new function: c_41(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2,u!`1!`1!`2) 555Integration of e_1 556 557Substitution of eta_u(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2,u!`1!`1!`2,u!`1!`1!`1) 558 by an expression in 559c_41(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2,u!`1!`1!`2), 560c_40(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2,u!`1!`1!`2) found in e_1 : 561eta_u= - (c_40 + c_41*u`1`1`1) 562 563 564Separation of e_2 yields (e_4 e_5) 565 566Separation of e_3 yields (e_6 e_7 e_9 e_8) 567 568new function: c_42(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2) 569Integration of e_4 570 571Substitution of c_41(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2,u!`1!`1!`2) by an 572expression in 573c_42(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2) found in e_4 : 574c_41= - c_42 575 576 577Separation of e_5 yields (e_10 e_11) 578 579new function: c_43(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2) 580new function: c_44(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2) 581Integration of e_7 582 583Substitution of c_40(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2,u!`1!`1!`2) by an 584expression in 585c_44(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2), 586c_43(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2) found in e_7 : 587c_40= - (c_43 + c_44*u`1`1`2) 588 589 590Separation of e_8 yields (e_12 e_13 e_14) 591 592Separation of e_9 yields (e_15 e_16 e_17 e_18) 593 594new function: c_45(t,x,u,u!`2,u!`1,u!`1!`2) 595Integration of e_10 596 597Substitution of c_42(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2) by an expression in 598c_45(t,x,u,u!`2,u!`1,u!`1!`2) found in e_10 : 599c_42= - c_45 600 601 602Separation of e_11 yields (e_19 e_20) 603 604new function: c_46(t,x,u,u!`2,u!`1,u!`1!`2) 605Integration of e_13 606 607Substitution of c_44(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2) by an expression in 608c_46(t,x,u,u!`2,u!`1,u!`1!`2) found in e_13 : 609c_44= - c_46 610 611 612Separation of e_14 yields (e_21 e_22 e_23) 613 614new function: c_47(t,x,u,u!`2,u!`1) 615Integration of e_19 616 617Substitution of c_45(t,x,u,u!`2,u!`1,u!`1!`2) by an expression in 618c_47(t,x,u,u!`2,u!`1) found in e_19 : 619c_45= - c_47 620 621 622Separation of e_20 yields (e_24 e_25) 623 624new function: c_48(t,x,u,u!`2,u!`1,u!`1!`2) 625new function: c_49(t,x,u,u!`2,u!`1,u!`1!`2) 626Integration of e_16 627 628Substitution of c_43(t,x,u,u!`2,u!`1,u!`1!`1,u!`1!`2) by an expression in 629c_49(t,x,u,u!`2,u!`1,u!`1!`2), c_48(t,x,u,u!`2,u!`1,u!`1!`2) found in e_16 : 630c_43= - (c_48 + c_49*u`1`1) 631 632 633Separation of e_17 yields (e_26 e_27 e_28) 634 635Separation of e_18 yields (e_29 e_30 e_31 e_32) 636 637new function: c_50(t,x,u,u!`2,u!`1) 638Integration of e_22 639 640Substitution of c_46(t,x,u,u!`2,u!`1,u!`1!`2) by an expression in 641c_50(t,x,u,u!`2,u!`1) found in e_22 : 642c_46= - c_50 643 644 645Separation of e_23 yields (e_33 e_34 e_35) 646 647new function: c_51(t,x,u,u!`2) 648Integration of e_24 649 650Substitution of c_47(t,x,u,u!`2,u!`1) by an expression in 651c_51(t,x,u,u!`2) found in e_24 : 652c_47= - c_51 653 654 655Separation of e_25 yields (e_36 e_37) 656 657new function: c_52(t,x,u,u!`2,u!`1) 658Integration of e_27 659 660Substitution of c_49(t,x,u,u!`2,u!`1,u!`1!`2) by an expression in 661c_52(t,x,u,u!`2,u!`1) found in e_27 : 662c_49= - c_52 663 664 665Separation of e_28 yields (e_38 e_39 e_40) 666 667new function: c_53(t,x,u,u!`2) 668Integration of e_34 669 670Substitution of c_50(t,x,u,u!`2,u!`1) by an expression in 671c_53(t,x,u,u!`2) found in e_34 : 672c_50= - c_53 673 674 675Separation of e_35 yields (e_41 e_42 e_43) 676 677new function: c_54(t,x,u) 678Integration of e_36 679 680Substitution of c_51(t,x,u,u!`2) by an expression in 681c_54(t,x,u) found in e_36 : 682c_51= - c_54 683 684 685Separation of e_37 yields (e_45 e_44) 686 687new function: c_55(t,x,u,u!`2,u!`1) 688new function: c_56(t,x,u,u!`2,u!`1) 689Integration of e_30 690 691Substitution of c_48(t,x,u,u!`2,u!`1,u!`1!`2) by an expression in 692c_56(t,x,u,u!`2,u!`1), c_55(t,x,u,u!`2,u!`1) found in e_30 : 693c_48= - (c_55 + c_56*u`1`2) 694 695 696Separation of e_31 yields (e_46 e_47 e_48) 697 698Separation of e_32 yields (e_49 e_50 e_52 e_51) 699 700new function: c_57(t,x,u) 701Integration of e_42 702 703Substitution of c_53(t,x,u,u!`2) by an expression in 704c_57(t,x,u) found in e_42 : 705c_53= - c_57 706 707 708Separation of e_43 yields (e_53 e_55 e_54) 709 710new function: c_58(t,u) 711Integration of e_45 712 713Substitution of c_54(t,x,u) by an expression in 714c_58(t,u) found in e_45 : 715c_54= - c_58 716 717 718new function: c_59(t,x,u,u!`2) 719new function: c_60(t,x,u,u!`2) 720Integration of e_49 721 722Substitution of c_56(t,x,u,u!`2,u!`1) by an expression in 723c_60(t,x,u,u!`2), c_59(t,x,u,u!`2) found in e_49 : 724c_56= - (c_59 + c_60*u`1) 725 726 727new function: c_61(t,x,u,u!`2) 728new function: c_62(t,x,u,u!`2) 729Integration of e_46 730 731Substitution of c_52(t,x,u,u!`2,u!`1) by an expression in 732c_62(t,x,u,u!`2), c_61(t,x,u,u!`2) found in e_46 : 733c_52= - (c_61 + c_62*u`1) 734 735 736Separation of e_47 yields (e_56 e_57) 737 738Separation of e_40 yields (e_58 e_59 e_60) 739 740e_58 is a consequence of e_56. 741Separation of e_48 yields (e_61 e_62 e_64 e_63) 742 743Substitution of c_62(t,x,u,u!`2) by an expression in 744c_58(t,u) found in e_39 : 745c_62=3*c_58 746 747 748new function: c_63(t,x) 749Integration of e_54 750 751Substitution of c_57(t,x,u) by an expression in 752c_63(t,x) found in e_54 : 753c_57= - c_63 754 755 756new function: c_64(t) 757Integration of e_44 758 759Substitution of c_58(t,u) by an expression in 760c_64(t) found in e_44 : 761c_58= - c_64 762 763 764Substitution of c_60(t,x,u,u!`2) by an expression in 765c_63(t,x) found in e_57 : 766c_60=2*c_63 767 768 769new function: c_65(t,x,u) 770new function: c_66(t,x,u) 771Integration of e_62 772 773Substitution of c_61(t,x,u,u!`2) by an expression in 774c_66(t,x,u), c_65(t,x,u) found in e_62 : 775c_61= - (c_65 + c_66*u`2) 776 777 778Separation of e_60 yields (e_3 e_2 e_1) 779 780Substitution of c_66(t,x,u) by an expression in 781c_63(t,x) found in e_59 : 782c_66= - c_63 783 784 785new function: c_67(t,x) 786Integration of e_2 787 788Substitution of c_65(t,x,u) by an expression in 789c_67(t,x) found in e_2 : 790c_65= - c_67 791 792 793new function: c_68(t) 794Integration of e_55 795 796Substitution of c_63(t,x) by an expression in 797c_64(t), c_68(t) found in e_55 : 798 - (c_64 *x + c_68) 799 t 800c_63=--------------------- 801 2 802 803 804new function: c_69(t,x,u) 805Integration of e_63 806 807Substitution of c_59(t,x,u,u!`2) by an expression in 808c_69(t,x,u) found in e_63 : 809c_59= - c_69 810 811 812Separation of e_64 yields (e_4 e_6) 813 814new function: c_70(t,x) 815new function: c_71(t,u) 816Partially potential integration of e_4 yields (e_4 e_5) 817 818Substitution of c_67(t,x) by an expression in 819c_70(t,x) found in e_5 : 820c_67=c_70 821 x 822 823 824Substitution of c_69(t,x,u) by an expression in 825c_71(t,u), c_70(t,x) found in e_4 : 826 - c_70 + c_70 - c_71 827 t 2x 828c_69=-------------------------- 829 2 830 831 832new function: c_72(t) 833new function: c_73(t) 834Integration of e_3 835 836Substitution of c_70(t,x) by an expression in 837c_64(t), c_68(t), c_73(t), c_72(t) found in e_3 : 838 3 2 839 - c_64 *x - 3*c_68 *x - 6*c_72 - 6*c_73*x 840 2t t 841c_70=----------------------------------------------- 842 24 843 844 845new function: c_74(t) 846Integration of e_6 847 848Substitution of c_71(t,u) by an expression in 849c_74(t) found in e_6 : 850c_71= - c_74 851 852 853new function: c_75(t) 854replacing df(c_72,t) by c_75(t) 855 856new function: c_76(t,x,u,u!`2) 857new function: c_77(t,x,u,u!`2) 858Integration of e_50 859 860Substitution of c_55(t,x,u,u!`2,u!`1) by an expression in 861c_77(t,x,u,u!`2), c_76(t,x,u,u!`2), c_64(t), c_68(t), c_73(t) found in e_50 : 28 862 factors in 9 terms 863 864Separation of e_51 yields (e_7 e_8) 865 866Separation of e_52 yields (e_9 e_12 e_11 e_10) 867 868new function: c_78(t,x,u) 869new function: c_79(t,x,u) 870Integration of e_9 871 872Substitution of c_77(t,x,u,u!`2) by an expression in 873c_79(t,x,u), c_78(t,x,u) found in e_9 : 874c_77= - (c_78 + c_79*u`2) 875 876 877Separation of e_8 yields (e_13 e_14 e_15) 878 879Substitution of c_79(t,x,u) by an expression in 880c_64(t), c_68(t), c_73(t), c_74(t), c_75(t) found in e_7 : 14 factors in 8 terms 881 882new function: c_80(t,x) 883Integration of e_13 884 885Substitution of c_78(t,x,u) by an expression in 886c_80(t,x) found in e_13 : 887c_78= - c_80 888 889 890new function: c_81(t,x,u) 891new function: c_82(t,x,u) 892Integration of e_12 893 894Substitution of c_76(t,x,u,u!`2) by an expression in 895c_82(t,x,u), c_81(t,x,u) found in e_12 : 896c_76= - (c_81 + c_82*u`2) 897 898 899Separation of e_11 yields (e_16 e_21 e_19 e_20) 900 901Separation of e_10 yields (e_18 e_17) 902 903new function: c_83(t,x) 904new function: c_84(t,x) 905Integration of e_21 906 907Substitution of c_82(t,x,u) by an expression in 908c_84(t,x), c_83(t,x) found in e_21 : 909 u 910c_82=e *c_83 - c_84 911 912 913Separation of e_18 yields (e_26 e_22) 914 915Separation of e_17 yields (e_23 e_33) 916 917e_33 is a consequence of e_14. 918Substitution of c_83(t,x) found in e_23 : 919c_83=0 920 921 922new function: c_85(t,x) 923new function: c_86(t,x) 924Integration of e_19 925 926Substitution of c_81(t,x,u) by an expression in 927c_86(t,x), c_85(t,x) found in e_19 : 928 u 929 - (e *c_86 + c_85) 930c_81=--------------------- 931 u 932 e 933 934 935Separation of e_16 yields (e_25 e_24) 936 937new function: c_87(t,x) 938Potential integration of e_20 yields (e_20 e_29) 939 940Substitution of c_84(t,x) by an expression in 941c_87(t,x) found in e_29 : 942c_84=c_87 943 x 944 945 946Substitution of c_86(t,x) by an expression in 947c_87(t,x) found in e_20 : 948 c_87 - c_87 949 t 2x 950c_86=---------------- 951 2 952 953 954new function: c_88(t) 955Integration of e_14 956 957Substitution of c_80(t,x) by an expression in 958c_64(t), c_68(t), c_73(t), c_74(t), c_75(t), c_88(t) found in e_14 : 22 959 factors in 9 terms 960 961new function: c_89(t) 962new function: c_90(t) 963Integration of e_22 964 965Substitution of c_87(t,x) by an expression in 966c_64(t), c_68(t), c_73(t), c_74(t), c_75(t), c_88(t), c_90(t), 967c_89(t) found in e_22 : 47 factors in 13 terms 968 969Separation of e_24 yields (e_28 e_27 e_38 e_35 e_34 e_36 e_41) 970 971new constant: c_91 972new constant: c_92 973new constant: c_93 974new constant: c_94 975new constant: c_95 976new constant: c_96 977Integration of e_28 978 979Substitution of c_68(t) by an expression in 980c_96, c_95, c_94, c_93, c_92, c_91 found in e_28 : 21 factors in 7 terms 981 982new constant: c_97 983new constant: c_98 984new constant: c_99 985new constant: c_100 986new constant: c_101 987new constant: c_102 988new constant: c_103 989Integration of e_27 990 991Substitution of c_64(t) by an expression in 992c_103, c_102, c_101, c_100, c_99, c_98, c_97 found in e_27 : 28 factors in 8 993 terms 994 995new function: c_104(t) 996replacing df(c_89,t) by c_104(t) 997 998new constant: c_105 999new constant: c_106 1000new constant: c_107 1001new constant: c_108 1002new constant: c_109 1003Integration of e_38 1004 1005Substitution of c_73(t) by an expression in 1006c_100, c_109, c_108, c_107, c_106, c_105 found in e_38 : 21 factors in 7 terms 1007 1008new constant: c_110 1009new constant: c_111 1010new constant: c_112 1011new constant: c_113 1012Integration of e_34 1013 1014Substitution of c_75(t) by an expression in 1015c_74(t), c_94, c_113, c_112, c_111, c_110 found in e_34 : 16 factors in 7 terms 1016 1017new constant: c_114 1018new constant: c_115 1019Integration of e_41 1020 1021Substitution of c_90(t) by an expression in 1022c_94, c_93, c_112, c_115, c_114 found in e_41 : 13 factors in 5 terms 1023 1024new constant: c_116 1025new constant: c_117 1026new constant: c_118 1027Integration of e_35 1028 1029Substitution of c_88(t) by an expression in 1030c_101, c_100, c_107, c_118, c_117, c_116 found in e_35 : 19 factors in 7 terms 1031 1032new constant: c_119 1033Integration of e_36 1034 1035Substitution of c_104(t) by an expression in 1036c_101, c_100, c_99, c_108, c_107, c_117, c_119 found in e_36 : 17 factors in 7 1037 terms 1038 1039 1040>>>>>>>>> Solution : 1041This is the end of the CRACK run 1042~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1043 1044-------- 1. Symmetry: 1045 1046xi_t=0 1047 1048xi_x=0 1049 1050eta_u=1 1051 1052-------- 2. Symmetry: 1053 1054xi_t=0 1055 1056xi_x=0 1057 1058eta_u=u 1059 t 1060 1061-------- 3. Symmetry: 1062 1063xi_t=0 1064 1065xi_x=0 1066 1067 2 2 1068eta_u=4*u *t + 4*u *t*x + 2*t + x 1069 t x 1070 1071-------- 4. Symmetry: 1072 1073xi_t=0 1074 1075xi_x=0 1076 1077eta_u=4*u *t + 2*u *x - 1 1078 t x 1079 1080-------- 5. Symmetry: 1081 1082xi_t=0 1083 1084xi_x=0 1085 1086eta_u=u 1087 x 1088 1089-------- 6. Symmetry: 1090 1091xi_t=0 1092 1093xi_x=0 1094 1095eta_u=2*u *t + x 1096 x 1097 1098-------- 7. Symmetry: 1099 1100xi_t=0 1101 1102xi_x=0 1103 1104eta_u=u + u *u 1105 t,x t x 1106 1107-------- 8. Symmetry: 1108 1109xi_t=0 1110 1111xi_x=0 1112 1113 2 2 2 1114eta_u=4*u *t + 4*u *u *t + 4*u *t*x + u *x - x 1115 t,x t x t x 1116 1117-------- 9. Symmetry: 1118 1119xi_t=0 1120 1121xi_x=0 1122 1123 3 3 2 2 2 3 1124eta_u=8*u *t + 8*u *u *t + 12*u *t *x + 12*u *t + 6*u *t*x + 6*t*x + x 1125 t,x t x t x x 1126 1127-------- 10. Symmetry: 1128 1129xi_t=0 1130 1131xi_x=0 1132 1133eta_u=2*u *t + 2*u *u *t + u *x 1134 t,x t x t 1135 1136-------- 11. Symmetry: 1137 1138xi_t=0 1139 1140xi_x=0 1141 1142 2 1143eta_u=u + u 1144 2t t 1145 1146-------- 12. Symmetry: 1147 1148xi_t=0 1149 1150xi_x=0 1151 1152 2 2 2 2 1153eta_u=8*u *t*x + 8*u *t + 8*u *t + 8*u *u *t*x + 2*u *x - 2*u *x + 1 1154 t,x 2t t t x t x 1155 1156-------- 13. Symmetry: 1157 1158xi_t=0 1159 1160xi_x=0 1161 1162 3 4 2 4 3 3 1163eta_u=32*u *t *x + 16*u *t + 16*u *t + 32*u *u *t *x + 48*u *t 1164 t,x 2t t t x t 1165 1166 2 2 2 3 2 2 4 1167 + 24*u *t *x + 48*u *t *x + 8*u *t*x + 12*t + 12*t*x + x 1168 t x x 1169 1170-------- 14. Symmetry: 1171 1172xi_t=0 1173 1174xi_x=0 1175 1176 2 3 2 3 2 2 1177eta_u=12*u *t *x + 8*u *t + 8*u *t + 12*u *u *t *x + 6*u *t*x - 6*u *t*x 1178 t,x 2t t t x t x 1179 1180 3 2 1181 + u *x - 6*t - 3*x 1182 x 1183 1184-------- 15. Symmetry: 1185 1186xi_t=0 1187 1188xi_x=0 1189 1190 2 1191eta_u=u *x + 2*u *t + 2*u *t + u *u *x 1192 t,x 2t t t x 1193 1194-------- 16. Symmetry: 1195 1196xi_t=0 1197 1198xi_x=0 1199 1200 3 1201eta_u=u + 3*u *u + u 1202 3t 2t t t 1203 1204-------- 17. Symmetry: 1205 1206xi_t=0 1207 1208xi_x=0 1209 1210 2 2 1211eta_u=8*u *u *t*x - u *x + 4*u *t + 4*u *t*x + 12*u *u *t 1212 t,x t t,x 3t 2t,x 2t t 1213 1214 2 3 2 2 2 2 1215 + 4*u *u *t*x + u *x + 4*u *t + 4*u *u *t*x + u *x - u *u *x 1216 2t x 2t t t x t t x 1217 1218-------- 18. Symmetry: 1219 1220xi_t=0 1221 1222xi_x=0 1223 1224 3 2 3 4 3 1225eta_u=64*u *u *t *x - 24*u *t *x + 8*u *t*x + 16*u *t + 32*u *t *x 1226 t,x t t,x t,x 3t 2t,x 1227 1228 4 3 2 2 3 4 2 3 1229 + 48*u *u *t + 32*u *u *t *x + 24*u *t *x + 16*u *t + 32*u *u *t *x 1230 2t t 2t x 2t t t x 1231 1232 2 2 2 2 3 2 4 1233 + 24*u *t *x - 24*u *u *t *x + 8*u *u *t*x - 24*u *t*x + u *x + 24*u *t*x 1234 t t x t x t t x 1235 1236 3 2 1237 - 6*u *x + 30*t + 15*x 1238 x 1239 1240-------- 19. Symmetry: 1241 1242xi_t=0 1243 1244xi_x=0 1245 1246 5 4 3 3 6 1247eta_u=384*u *u *t *x + 960*u *t *x + 160*u *t *x + 64*u *t 1248 t,x t t,x t,x 3t 1249 1250 5 6 5 5 4 2 1251 + 192*u *t *x + 192*u *u *t + 192*u *u *t *x + 480*u *t + 240*u *t *x 1252 2t,x 2t t 2t x 2t 2t 1253 1254 3 6 2 5 2 5 2 4 2 4 1255 + 64*u *t + 192*u *u *t *x + 480*u *t + 240*u *t *x + 960*u *u *t *x 1256 t t x t t t x 1257 1258 3 3 4 3 2 2 4 3 1259 + 160*u *u *t *x + 720*u *t + 720*u *t *x + 60*u *t *x + 720*u *t *x 1260 t x t t t x 1261 1262 2 3 5 3 2 2 4 6 1263 + 240*u *t *x + 12*u *t*x + 120*t + 180*t *x + 30*t*x + x 1264 x x 1265 1266-------- 20. Symmetry: 1267 1268xi_t=0 1269 1270xi_x=0 1271 1272 4 3 2 3 5 1273eta_u=160*u *u *t *x - 80*u *t *x + 40*u *t *x + 32*u *t 1274 t,x t t,x t,x 3t 1275 1276 4 5 4 3 2 3 5 1277 + 80*u *t *x + 96*u *u *t + 80*u *u *t *x + 80*u *t *x + 32*u *t 1278 2t,x 2t t 2t x 2t t 1279 1280 2 4 2 3 2 3 2 3 3 1281 + 80*u *u *t *x + 80*u *t *x - 80*u *u *t *x + 40*u *u *t *x - 360*u *t 1282 t x t t x t x t 1283 1284 2 2 4 2 3 5 2 1285 - 120*u *t *x + 10*u *t*x - 420*u *t *x - 60*u *t*x + u *x - 120*t 1286 t t x x x 1287 1288 2 4 1289 - 120*t*x - 10*x 1290 1291-------- 21. Symmetry: 1292 1293xi_t=0 1294 1295xi_x=0 1296 1297 2 3 3 2 1298eta_u=48*u *u *t *x - 12*u *t*x + 2*u *x + 16*u *t + 24*u *t *x 1299 t,x t t,x t,x 3t 2t,x 1300 1301 3 2 2 3 3 2 2 1302 + 48*u *u *t + 24*u *u *t *x + 12*u *t*x + 16*u *t + 24*u *u *t *x 1303 2t t 2t x 2t t t x 1304 1305 2 2 3 2 1306 + 12*u *t*x - 12*u *u *t*x + 2*u *u *x - 6*u *x + 6*u *x - 3 1307 t t x t x t x 1308 1309-------- 22. Symmetry: 1310 1311xi_t=0 1312 1313xi_x=0 1314 1315eta_u 1316 1317 3 2 1318=2*u *u *x + 2*u *t + u *x + 6*u *u *t + u *u *x + 2*u *t + u *u *x 1319 t,x t 3t 2t,x 2t t 2t x t t x 1320 1321-------- 23. Symmetry: 1322 1323xi_t=0 1324 1325xi_x=0 1326 1327 2 1328eta_u=2*u *u + u + u *u + u *u 1329 t,x t 2t,x 2t x t x 1330 1331-------- 24. Symmetry: 1332 1333xi_t=0 1334 1335xi_x=0 1336 1337 2 2 2 2 1338eta_u=8*u *u *t - 2*u *t + u *x + 4*u *t + 4*u *u *t + 4*u *t*x 1339 t,x t t,x t,x 2t,x 2t x 2t 1340 1341 2 2 2 2 1342 + 4*u *u *t + 4*u *t*x - 2*u *u *t + u *u *x - 2*u *x 1343 t x t t x t x t 1344 1345-------- 25. Symmetry: 1346 1347xi_t=0 1348 1349xi_x=0 1350 1351 4 3 2 2 4 4 1352eta_u=32*u *u *t - 16*u *t + 24*u *t *x + 16*u *t + 16*u *u *t 1353 t,x t t,x t,x 2t,x 2t x 1354 1355 3 2 4 2 3 3 2 2 1356 + 32*u *t *x + 16*u *u *t + 32*u *t *x - 16*u *u *t + 24*u *u *t *x 1357 2t t x t t x t x 1358 1359 2 3 2 2 4 3 1360 - 48*u *t *x + 8*u *t*x - 84*u *t - 36*u *t*x + u *x - 48*t*x - 8*x 1361 t t x x x 1362 1363-------- 26. Symmetry: 1364 1365xi_t=0 1366 1367xi_x=0 1368 1369 5 4 3 2 5 5 1370eta_u=64*u *u *t + 160*u *t + 80*u *t *x + 32*u *t + 32*u *u *t 1371 t,x t t,x t,x 2t,x 2t x 1372 1373 4 2 5 2 4 4 3 2 1374 + 80*u *t *x + 32*u *u *t + 80*u *t *x + 160*u *u *t + 80*u *u *t *x 1375 2t t x t t x t x 1376 1377 3 2 3 3 2 2 4 2 1378 + 240*u *t *x + 40*u *t *x + 120*u *t + 120*u *t *x + 10*u *t*x + 60*t *x 1379 t t x x x 1380 1381 3 5 1382 + 20*t*x + x 1383 1384-------- 27. Symmetry: 1385 1386xi_t=0 1387 1388xi_x=0 1389 1390 3 2 2 3 3 1391eta_u=32*u *u *t - 12*u *t + 12*u *t*x + 16*u *t + 16*u *u *t 1392 t,x t t,x t,x 2t,x 2t x 1393 1394 2 2 3 2 2 2 2 1395 + 24*u *t *x + 16*u *u *t + 24*u *t *x - 12*u *u *t + 12*u *u *t*x 1396 2t t x t t x t x 1397 1398 3 2 1399 - 24*u *t*x + 2*u *x - 9*u *x + 9*x 1400 t t x 1401 1402-------- 28. Symmetry: 1403 1404xi_t=0 1405 1406xi_x=0 1407 1408 2 1409eta_u=8*u *u *t - u + 4*u *t + 4*u *u *t + 2*u *x + 4*u *u *t 1410 t,x t t,x 2t,x 2t x 2t t x 1411 1412 2 1413 + 2*u *x - u *u 1414 t t x 1415 1416-------- Further symmetry: 1417 1418xi_t=0 1419 1420xi_x=0 1421 1422 c_85 1423eta_u=------ 1424 u 1425 e 1426 1427with c_85(t,x) 1428which still has to satisfy: 1429 14300=c_85 - c_85 1431 t 2x 1432 1433 1434-------- 1435 1436=============================================================== 1437 1438-------------------------------------------------------------------------- 1439 1440This is LIEPDE - a program for calculating infinitesimal symmetries 1441of single differential equations or systems of de's 1442 1443The PDE-system under investigation is : 1444 1445 2 2 2 2 2 2 2 1446v =( - 4*f *a2*r - 2*f *a2*r *s1 - 2*f *a2*r *s1 - 2*f *a2*r *s2 1447 2x t x y z 1448 1449 2 2 2 1450 - 4*f *a2*r *w1 - 2*r *a2*s1 - 2*r *a2*s1 - 2*r *a2*r*s1 1451 z x y 2z 1452 1453 2 2 2 1454 + 2*r *a2*r*s2 - 2*r *a2*s1 + v *s1 - v *s1*w2 - v *s1*w2 1455 2z z 2t 2y 2z 1456 1457 2 2 1458 - 4*a1*a2*r *v)/(s1*w2 ) 1459 1460 2 2 2 1461r =(2*f *r + f *r*s1 + f *r*s1 + f *r*s2 + 2*f *r*w1 - r *s1 - r *s2 1462 2x t x y z z 2y 2z 1463 1464 + 2*a1*r*v)/s1 1465 1466f = 1467 2x 1468 1469 - 2*f *r *s1 - f *r*s1 - 2*f *r *s1 - f *r*s2 - 2*f *r *s2 - 2*r - 2*r *w1 1470 x x 2y y y 2z z z t z 1471-------------------------------------------------------------------------------- 1472 r*s1 1473 1474for the function(s) : 1475 1476r(z,y,t,x), f(z,y,t,x), v(z,y,t,x) 1477 1478 1479Start of splitting equations 14805 equations result 1481 1482Start of splitting equations 14837 equations result 1484 1485Start of splitting equations 14867 equations result 1487 1488Start of splitting equations 14894 equations result 1490 1491Start of splitting equations 14927 equations result 1493 1494Start of splitting equations 14954 equations result 1496 1497Start of splitting equations 14984 equations result 1499 1500Start of splitting equations 15015 equations result 1502 1503Start of splitting equations 15045 equations result 1505 1506~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1507This is CRACK - a solver for overdetermined partial differential equations 1508 1509e_6 is a consequence of e_1. 1510e_7 is a consequence of e_2. 1511e_8 is a consequence of e_3. 1512e_9 is a consequence of e_4. 1513e_13 is a consequence of e_4. 1514e_15 is a consequence of e_11. 1515e_16 is a consequence of e_12. 1516e_17 is a consequence of e_4. 1517e_23 is a consequence of e_3. 1518e_24 is a consequence of e_2. 1519e_26 is a consequence of e_11. 1520e_27 is a consequence of e_12. 1521e_28 is a consequence of e_4. 1522e_30 is a consequence of e_19. 1523e_32 is a consequence of e_21. 1524e_34 is a consequence of e_3. 1525e_35 is a consequence of e_1. 1526e_37 is a consequence of e_20. 1527e_38 is a consequence of e_31. 1528e_39 is a consequence of e_22. 1529e_40 is a consequence of e_33. 1530e_41 is a consequence of e_2. 1531e_42 is a consequence of e_1. 1532e_44 is a consequence of e_1. 1533e_45 is a consequence of e_2. 1534e_46 is a consequence of e_3. 1535e_47 is a consequence of e_4. 1536new function: c_120(y,z,t,v,f,r) 1537Integration of e_29 1538 1539Substitution of xi_t(x,y,z,t,v,f,r) by an expression in 1540c_120(y,z,t,v,f,r) found in e_29 : 1541xi_t= - c_120 1542 1543 1544new function: c_121(z,t,v,f,r) 1545Integration of e_18 1546 1547Substitution of c_120(y,z,t,v,f,r) by an expression in 1548c_121(z,t,v,f,r) found in e_18 : 1549c_120= - c_121 1550 1551 1552new function: c_122(t,v,f,r) 1553Integration of e_14 1554 1555Substitution of c_121(z,t,v,f,r) by an expression in 1556c_122(t,v,f,r) found in e_14 : 1557c_121= - c_122 1558 1559 1560new function: c_123(t,f,r) 1561Integration of e_4 1562 1563Substitution of c_122(t,v,f,r) by an expression in 1564c_123(t,f,r) found in e_4 : 1565c_122= - c_123 1566 1567 1568new function: c_124(x,y,z,t,f,r) 1569Integration of e_3 1570 1571Substitution of xi_z(x,y,z,t,v,f,r) by an expression in 1572c_124(x,y,z,t,f,r) found in e_3 : 1573xi_z= - c_124 1574 1575 1576new function: c_125(x,y,z,t,f,r) 1577Integration of e_2 1578 1579Substitution of xi_y(x,y,z,t,v,f,r) by an expression in 1580c_125(x,y,z,t,f,r) found in e_2 : 1581xi_y= - c_125 1582 1583 1584new function: c_126(x,y,z,t,f,r) 1585Integration of e_1 1586 1587Substitution of xi_x(x,y,z,t,v,f,r) by an expression in 1588c_126(x,y,z,t,f,r) found in e_1 : 1589xi_x= - c_126 1590 1591 1592new function: c_127(x,y,z,t,r) 1593Integration of e_31 1594 1595Substitution of c_126(x,y,z,t,f,r) by an expression in 1596c_127(x,y,z,t,r) found in e_31 : 1597c_126= - c_127 1598 1599 1600new function: c_128(x,y,z,t,r) 1601Integration of e_20 1602 1603Substitution of c_125(x,y,z,t,f,r) by an expression in 1604c_128(x,y,z,t,r) found in e_20 : 1605c_125= - c_128 1606 1607 1608new function: c_129(x,y,z,t,r) 1609Integration of e_19 1610 1611Substitution of c_124(x,y,z,t,f,r) by an expression in 1612c_129(x,y,z,t,r) found in e_19 : 1613c_124= - c_129 1614 1615 1616new function: c_130(t,r) 1617Integration of e_11 1618 1619Substitution of c_123(t,f,r) by an expression in 1620c_130(t,r) found in e_11 : 1621c_123= - c_130 1622 1623 1624new function: c_131(t) 1625Integration of e_12 1626 1627Substitution of c_130(t,r) by an expression in 1628c_131(t) found in e_12 : 1629c_130= - c_131 1630 1631 1632new function: c_132(x,y,z,t) 1633Integration of e_21 1634 1635Substitution of c_129(x,y,z,t,r) by an expression in 1636c_132(x,y,z,t) found in e_21 : 1637c_129= - c_132 1638 1639 1640new function: c_133(x,y,z,t) 1641Integration of e_22 1642 1643Substitution of c_128(x,y,z,t,r) by an expression in 1644c_133(x,y,z,t) found in e_22 : 1645c_128= - c_133 1646 1647 1648new function: c_134(x,y,z,t) 1649Integration of e_33 1650 1651Substitution of c_127(x,y,z,t,r) by an expression in 1652c_134(x,y,z,t) found in e_33 : 1653c_127= - c_134 1654 1655 1656From now on lexicographic ordering of derivatives. 1657The current variable ordering is going to be reversed. 1658 1659>>>>>>>>> Solution : 1660This is the end of the CRACK run 1661~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1662 1663Remaining free functions after the last CRACK-run: 1664eta_r(r,f,v,t,z,y,x), eta_f(r,f,v,t,z,y,x), eta_v(r,f,v,t,z,y,x), 1665c_132(t,z,y,x), c_133(t,z,y,x), c_134(t,z,y,x), c_131(t) 1666 1667 1668Start of splitting equations 16691 equations result 1670 1671Start of splitting equations 16721 equations result 1673 1674Start of splitting equations 16751 equations result 1676 1677~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1678This is CRACK - a solver for overdetermined partial differential equations 1679 1680e_4 is a consequence of e_3. 1681e_5 is a consequence of e_2. 1682e_6 is a consequence of e_1. 1683The current variable ordering is going to be reversed. 1684 1685>>>>>>>>> Solution : 1686This is the end of the CRACK run 1687~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1688 1689Remaining free functions after the last CRACK-run: 1690c_131(t), c_134(t,z,y,x), c_133(t,z,y,x), c_132(t,z,y,x), 1691eta_v(r,f,v,t,z,y,x), eta_f(r,f,v,t,z,y,x), eta_r(r,f,v,t,z,y,x) 1692 1693 1694Start of splitting equations 16951 equations result 1696 1697Start of splitting equations 16981 equations result 1699 1700Start of splitting equations 17011 equations result 1702 1703Start of splitting equations 17041 equations result 1705 1706Start of splitting equations 17071 equations result 1708 1709Start of splitting equations 17101 equations result 1711 1712Start of splitting equations 17131 equations result 1714 1715Start of splitting equations 17161 equations result 1717 1718~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1719This is CRACK - a solver for overdetermined partial differential equations 1720 1721e_9 is a consequence of e_6. 1722new function: c_135(x,y,t) 1723Integration of e_8 1724 1725e_10 is a consequence of e_5. 1726Substitution of c_134(x,y,z,t) by an expression in 1727c_135(x,y,t) found in e_8 : 1728c_134= - c_135 1729 1730 1731new function: c_136(x,y,t) 1732Integration of e_7 1733 1734e_11 is a consequence of e_3. 1735Substitution of c_133(x,y,z,t) by an expression in 1736c_136(x,y,t) found in e_7 : 1737c_133= - c_136 1738 1739 1740new function: c_137(y,z,t) 1741Integration of e_5 1742 1743Substitution of c_132(x,y,z,t) by an expression in 1744c_137(y,z,t) found in e_5 : 1745c_132= - c_137 1746 1747 1748new function: c_138(z,t) 1749Integration of e_3 1750 1751Substitution of c_137(y,z,t) by an expression in 1752c_138(z,t) found in e_3 : 1753c_137= - c_138 1754 1755 1756new function: c_139(z) 1757Integration of e_1 1758 1759Substitution of c_138(z,t) by an expression in 1760c_139(z) found in e_1 : 1761c_138= - c_139 1762 1763 1764new function: c_140(x,y) 1765Integration of e_2 1766 1767Substitution of c_136(x,y,t) by an expression in 1768c_140(x,y) found in e_2 : 1769c_136= - c_140 1770 1771 1772new function: c_141(x,y) 1773Integration of e_4 1774 1775Substitution of c_135(x,y,t) by an expression in 1776c_141(x,y) found in e_4 : 1777c_135= - c_141 1778 1779 1780 1781>>>>>>>>> Solution : 1782This is the end of the CRACK run 1783~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1784 1785Remaining free functions after the last CRACK-run: 1786c_131(t), eta_r(x,y,z,t,v,f,r), eta_f(x,y,z,t,v,f,r), 1787eta_v(x,y,z,t,v,f,r), c_140(x,y), c_141(x,y), c_139(z) 1788 1789 1790Start of splitting equations 179139 equations result 1792 1793Start of splitting equations 179437 equations result 1795 1796Start of splitting equations 179747 equations result 1798 1799~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1800This is CRACK - a solver for overdetermined partial differential equations 1801 1802e_7 is a consequence of e_3. 1803e_8 is a consequence of e_4. 1804e_21 is a consequence of e_2. 1805e_25 is a consequence of e_22. 1806e_37 is a consequence of e_2. 1807e_38 is a consequence of e_35. 1808e_44 is a consequence of e_35. 1809e_49 is a consequence of e_2. 1810e_54 is a consequence of e_50. 1811e_55 is a consequence of e_51. 1812e_58 is a consequence of e_50. 1813e_59 is a consequence of e_51. 1814e_65 is a consequence of e_2. 1815e_66 is a consequence of e_62. 1816e_67 is a consequence of e_63. 1817e_69 is a consequence of e_40. 1818e_71 is a consequence of e_63. 1819e_77 is a consequence of e_75. 1820e_79 is a consequence of e_75. 1821e_83 is a consequence of e_81. 1822e_87 is a consequence of e_2. 1823e_90 is a consequence of e_2. 1824e_91 is a consequence of e_85. 1825e_92 is a consequence of e_2. 1826e_94 is a consequence of e_88. 1827e_96 is a consequence of e_40. 1828e_97 is a consequence of e_85. 1829e_99 is a consequence of e_88. 1830e_106 is a consequence of e_103. 1831e_107 is a consequence of e_104. 1832e_110 is a consequence of e_104. 1833e_112 is a consequence of e_82. 1834e_116 is a consequence of e_114. 1835e_118 is a consequence of e_114. 1836e_120 is a consequence of e_82. 1837e_121 is a consequence of e_81. 1838e_122 is a consequence of e_82. 1839e_124 is a consequence of e_2. 1840new function: c_142(x,y,z,t,f,r) 1841Integration of e_82 1842 1843e_4 is a consequence of e_17. 1844e_119 is a consequence of e_81. 1845e_88 is a consequence of e_81. 1846e_102 is a consequence of e_74. 1847e_103 is a consequence of e_109. 1848Substitution of eta_f(x,y,z,t,v,f,r) by an expression in 1849c_142(x,y,z,t,f,r) found in e_82 : 1850eta_f= - c_142 1851 1852 1853new function: c_143(x,y,z,t,f,r) 1854Integration of e_81 1855 1856e_22 is a consequence of e_32. 1857e_70 is a consequence of e_62. 1858e_41 is a consequence of e_35. 1859e_12 is a consequence of e_17. 1860e_28 is a consequence of e_32. 1861e_73 is a consequence of e_101. 1862e_43 is a consequence of e_101. 1863Substitution of eta_r(x,y,z,t,v,f,r) by an expression in 1864c_143(x,y,z,t,f,r) found in e_81 : 1865eta_r= - c_143 1866 1867 1868Separation of e_123 yields (e_7 e_8) 1869 1870new function: c_144(x,y,z,t,v,r) 1871Integration of e_19 1872 1873e_3 is a consequence of e_74. 1874Substitution of eta_v(x,y,z,t,v,f,r) by an expression in 1875c_144(x,y,z,t,v,r) found in e_19 : 1876eta_v= - c_144 1877 1878 1879new function: c_145(x,y,z,t,f) 1880Integration of e_7 1881 1882e_74 is a consequence of e_14. 1883e_105 is a consequence of e_5. 1884e_108 is a consequence of e_9. 1885Substitution of c_142(x,y,z,t,f,r) by an expression in 1886c_145(x,y,z,t,f) found in e_7 : 1887c_142= - c_145 1888 1889 1890new function: c_146(x,y,z,t,r) 1891new function: c_147(x,y,z,t,r) 1892Integration of e_35 1893 1894Substitution of c_144(x,y,z,t,v,r) by an expression in 1895c_147(x,y,z,t,r), c_146(x,y,z,t,r) found in e_35 : 1896c_144= - (c_146 + c_147*v) 1897 1898 1899Separation of e_33 yields (e_25 e_21) 1900 1901Separation of e_31 yields (e_38 e_37) 1902 1903Separation of e_34 yields (e_49 e_44) 1904 1905e_44 is a consequence of e_32. 1906Separation of e_26 yields (e_54 e_55) 1907 1908Separation of e_23 yields (e_58 e_59) 1909 1910Separation of e_20 yields (e_65 e_66) 1911 1912e_65 is a consequence of e_37. 1913Separation of e_1 yields (e_67 e_69) 1914 1915e_67 is a consequence of e_32. 1916Separation of e_24 yields (e_71 e_77) 1917 1918e_71 is a consequence of e_37. 1919Separation of e_30 yields (e_79 e_83) 1920 1921e_79 is a consequence of e_32. 1922Separation of e_18 yields (e_87 e_90) 1923 1924e_87 is a consequence of e_32. 1925Separation of e_6 yields (e_91 e_92) 1926 1927e_91 is a consequence of e_32. 1928Separation of e_29 yields (e_94 e_96) 1929 1930Separation of e_27 yields (e_97 e_99) 1931 1932e_97 is a consequence of e_37. 1933Separation of e_10 yields (e_106 e_107) 1934 1935e_106 is a consequence of e_32. 1936Separation of e_84 yields (e_110 e_112) 1937 1938Separation of e_13 yields (e_116 e_118) 1939 1940e_116 is a consequence of e_32. 1941Separation of e_47 yields (e_120 e_121 e_122) 1942 1943e_120 is a consequence of e_32. 1944new function: c_148(x,y,z,t,r) 1945Integration of e_14 1946 1947e_52 is a consequence of e_5. 1948e_56 is a consequence of e_9. 1949Substitution of c_143(x,y,z,t,f,r) by an expression in 1950c_148(x,y,z,t,r) found in e_14 : 1951c_143= - c_148 1952 1953 1954new function: c_149(y,z,t,f) 1955Integration of e_5 1956 1957Substitution of c_145(x,y,z,t,f) by an expression in 1958c_149(y,z,t,f) found in e_5 : 1959c_145= - c_149 1960 1961 1962new function: c_150(x,y,z,t) 1963Integration of e_49 1964 1965Substitution of c_146(x,y,z,t,r) by an expression in 1966c_150(x,y,z,t) found in e_49 : 1967c_146= - c_150 1968 1969 1970new function: c_151(x,y,z,t) 1971Integration of e_32 1972 1973Substitution of c_147(x,y,z,t,r) by an expression in 1974c_151(x,y,z,t) found in e_32 : 1975c_147= - c_151 1976 1977 1978new function: c_152(y,z,t) 1979new function: c_153(y,z,t) 1980Integration of e_85 1981 1982Substitution of c_149(y,z,t,f) by an expression in 1983c_153(y,z,t), c_152(y,z,t) found in e_85 : 1984c_149= - (c_152 + c_153*f) 1985 1986 1987Separation of e_9 yields (e_117 e_124) 1988 1989Separation of e_8 yields (e_115 e_114) 1990 1991Separation of e_60 yields (e_4 e_119) 1992 1993Separation of e_111 yields (e_104 e_88) 1994 1995e_104 is a consequence of e_4. 1996Separation of e_118 yields (e_102 e_103) 1997 1998e_102 is a consequence of e_4. 1999Separation of e_112 yields (e_82 e_51) 2000 2001Separation of e_122 yields (e_80 e_78) 2002 2003e_80 is a consequence of e_82. 2004new function: c_154(x,y,z,t) 2005new function: c_155(x,y,z,t) 2006Integration of e_62 2007 2008Substitution of c_148(x,y,z,t,r) by an expression in 2009c_155(x,y,z,t), c_154(x,y,z,t) found in e_62 : 2010c_148= - (c_154 + c_155*r) 2011 2012 2013Separation of e_59 yields (e_75 e_76) 2014 2015Separation of e_55 yields (e_22 e_63) 2016 2017Separation of e_48 yields (e_41 e_70) 2018 2019e_41 is a consequence of e_86. 2020Separation of e_98 yields (e_28 e_12) 2021 2022e_28 is a consequence of e_86. 2023e_12 is a consequence of e_101. 2024Separation of e_93 yields (e_43 e_73) 2025 2026e_43 is a consequence of e_86. 2027e_73 is a consequence of e_40. 2028Separation of e_110 yields (e_123 e_81) 2029 2030e_123 is a consequence of e_86. 2031Separation of e_113 yields (e_16 e_17) 2032 2033e_16 is a consequence of e_86. 2034Separation of e_61 yields (e_3 e_15) 2035 2036e_3 is a consequence of e_86. 2037Separation of e_53 yields (e_74 e_19) 2038 2039e_74 is a consequence of e_86. 2040Separation of e_57 yields (e_105 e_109) 2041 2042e_105 is a consequence of e_86. 2043Separation of e_89 yields (e_7 e_108) 2044 2045e_7 is a consequence of e_75. 2046e_108 is a consequence of e_64. 2047Separation of e_100 yields (e_33 e_35) 2048 2049Separation of e_69 yields (e_34 e_31) 2050 2051e_34 is a consequence of e_86. 2052Separation of e_88 yields (e_26 e_44) 2053 2054e_26 is a consequence of e_86. 2055Separation of e_119 yields (e_20 e_23) 2056 2057e_20 is a consequence of e_86. 2058Separation of e_83 yields (e_1 e_65) 2059 2060e_1 is a consequence of e_86. 2061e_65 is a consequence of e_99. 2062Separation of e_95 yields (e_24 e_67) 2063 2064e_24 is a consequence of e_22. 2065Separation of e_96 yields (e_30 e_71) 2066 2067e_30 is a consequence of e_33. 2068Separation of e_92 yields (e_18 e_79) 2069 2070e_18 is a consequence of e_86. 2071Separation of e_90 yields (e_6 e_87) 2072 2073e_6 is a consequence of e_86. 2074Separation of e_107 yields (e_29 e_91) 2075 2076e_29 is a consequence of e_86. 2077Separation of e_114 yields (e_97 e_27) 2078 2079Separation of e_121 yields (e_106 e_10 e_84) 2080 2081e_106 is a consequence of e_86. 2082Separation of e_103 yields (e_116 e_13) 2083 2084e_116 is a consequence of e_86. 2085Separation of e_51 yields (e_120 e_47) 2086 2087Separation of e_78 yields (e_56 e_52 e_50) 2088 2089Substitution of c_154(x,y,z,t) found in e_86 : 2090c_154=0 2091 2092 2093e_66 is a consequence of e_10. 2094Substitution of c_151(x,y,z,t) by an expression in 2095c_141(x,y) found in e_81 : 2096c_151= - 2*c_141 2097 x 2098 2099 2100e_72 is a consequence of e_42. 2101e_77 is a consequence of e_40. 2102e_99 is a consequence of e_101. 2103e_31 is a consequence of e_70. 2104e_79 is a consequence of e_19. 2105e_87 is a consequence of e_15. 2106e_91 is a consequence of e_109. 2107e_71 is a consequence of e_42. 2108e_13 is a consequence of e_23. 2109Substitution of c_155(x,y,z,t) by an expression in 2110c_141(x,y) found in e_10 : 2111c_155= - c_141 2112 x 2113 2114 2115e_35 is a consequence of e_42. 2116e_15 is a consequence of e_17. 2117e_19 is a consequence of e_40. 2118e_109 is a consequence of e_101. 2119e_23 is a consequence of e_44. 2120e_67 is a consequence of e_68. 2121Substitution of c_153(y,z,t) found in e_70 : 2122c_153=0 2123 2124 2125new function: c_156(z,t) 2126Integration of e_117 2127 2128Substitution of c_152(y,z,t) by an expression in 2129c_156(z,t) found in e_117 : 2130c_152= - c_156 2131 2132 2133new function: c_157(y) 2134new function: c_158(y) 2135Integration of e_76 2136 2137e_64 is a consequence of e_36. 2138Substitution of c_141(x,y) by an expression in 2139c_158(y), c_157(y) found in e_76 : 2140c_141= - (c_157 + c_158*x) 2141 2142 2143Separation of e_36 yields (e_14 e_11) 2144 2145e_11 is a consequence of e_84. 2146new function: c_159(t) 2147new function: c_160(t) 2148Integration of e_27 2149 2150Substitution of c_156(z,t) by an expression in 2151c_160(t), c_159(t) found in e_27 : 2152c_156= - (c_159 + c_160*z) 2153 2154 2155new constant: c_161 2156new constant: c_162 2157Integration of e_42 2158 2159Substitution of c_139(z) by an expression in 2160c_162, c_161 found in e_42 : 2161c_139= - (c_161 + c_162*z) 2162 2163 2164e_39 is a consequence of e_68. 2165e_47 is a consequence of e_52. 2166Substitution of c_158(y) by an expression in 2167c_162 found in e_101 : 2168c_158= - c_162 2169 2170 2171Substitution of c_160(t) by an expression in 2172c_162 found in e_44 : 2173 c_162*w1 2174c_160=---------- 2175 s2 2176 2177 2178new constant: c_163 2179new constant: c_164 2180Integration of e_14 2181 2182Substitution of c_157(y) by an expression in 2183c_164, c_163 found in e_14 : 2184c_157= - (c_163 + c_164*y) 2185 2186 2187new constant: c_165 2188new constant: c_166 2189Integration of e_45 2190 2191Substitution of c_131(t) by an expression in 2192c_166, c_165 found in e_45 : 2193c_131= - (c_165 + c_166*t) 2194 2195 2196Substitution of c_166 by an expression in 2197c_162 found in e_17 : 2198c_166=2*c_162 2199 2200 2201Substitution of c_162 found in e_46 : 2202c_162=0 2203 2204 2205Substitution of c_150(x,y,z,t) by an expression in 2206c_159(t) found in e_52 : 2207 c_159 2208 t 2209c_150=-------- 2210 a1 2211 2212 2213new function: c_167(x) 2214Integration of e_40 2215 2216Substitution of c_140(x,y) by an expression in 2217c_167(x) found in e_40 : 2218c_140= - c_167 2219 2220 2221new constant: c_168 2222new constant: c_169 2223Integration of e_68 2224 2225Substitution of c_167(x) by an expression in 2226c_169, c_168 found in e_68 : 2227c_167= - (c_168 + c_169*x) 2228 2229 2230Substitution of c_164 by an expression in 2231c_169 found in e_2 : 2232c_164= - c_169 2233 2234 2235new constant: c_170 2236new constant: c_171 2237new constant: c_172 2238Integration of e_50 2239 2240Substitution of c_159(t) by an expression in 2241c_172, c_171, c_170 found in e_50 : 2242 2 2243 - 2*c_170 - c_171*t - 2*c_172*t 2244c_159=----------------------------------- 2245 2 2246 2247 2248 2249>>>>>>>>> Solution : 2250This is the end of the CRACK run 2251~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2252 2253-------- 1. Symmetry: 2254 2255xi_x=0 2256 2257xi_y=0 2258 2259xi_z=0 2260 2261xi_t=0 2262 2263eta_r=0 2264 2265eta_f=1 2266 2267eta_v=0 2268 2269-------- 2. Symmetry: 2270 2271xi_x=0 2272 2273xi_y=0 2274 2275xi_z=0 2276 2277xi_t=0 2278 2279eta_r=0 2280 2281 2 2282eta_f= - t 2283 2284 2*t 2285eta_v=----- 2286 a1 2287 2288-------- 3. Symmetry: 2289 2290xi_x=0 2291 2292xi_y=0 2293 2294xi_z=0 2295 2296xi_t=0 2297 2298eta_r=0 2299 2300eta_f= - t 2301 2302 1 2303eta_v=---- 2304 a1 2305 2306-------- 4. Symmetry: 2307 2308xi_x=0 2309 2310xi_y=1 2311 2312xi_z=0 2313 2314xi_t=0 2315 2316eta_r=0 2317 2318eta_f=0 2319 2320eta_v=0 2321 2322-------- 5. Symmetry: 2323 2324xi_x= - y 2325 2326xi_y=x 2327 2328xi_z=0 2329 2330xi_t=0 2331 2332eta_r=0 2333 2334eta_f=0 2335 2336eta_v=0 2337 2338-------- 6. Symmetry: 2339 2340xi_x=0 2341 2342xi_y=0 2343 2344xi_z=0 2345 2346xi_t=1 2347 2348eta_r=0 2349 2350eta_f=0 2351 2352eta_v=0 2353 2354-------- 7. Symmetry: 2355 2356xi_x=1 2357 2358xi_y=0 2359 2360xi_z=0 2361 2362xi_t=0 2363 2364eta_r=0 2365 2366eta_f=0 2367 2368eta_v=0 2369 2370-------- 8. Symmetry: 2371 2372xi_x=0 2373 2374xi_y=0 2375 2376xi_z=1 2377 2378xi_t=0 2379 2380eta_r=0 2381 2382eta_f=0 2383 2384eta_v=0 2385 2386 2387-------- 2388 2389=============================================================== 2390Your ansatz for the symmetry needs to be homogeneous, i.e. 2391substituting all unknown functions and constants to be computed 2392(which do not occur in the equation) to zero needs to make the 2393symmetry to zero. In your ansatz this is not 2394the case because the list of substitutions: 2395{g=0,f=0} 2396leaves this right hand side non-zero: 2397eta_u=u 2398 4x 2399To fix your ansatz you could, for example, simply multiply all 2400non-vanishing parts on all right hand sides in your ansatz with one 2401and the same unknown constant, say cc, and add cc to the list of unknowns 2402to be computed and to the list of non-vanishing expressions. 2403Your ansatz for the symmetry needs to be homogeneous, i.e. 2404substituting all unknown functions and constants to be computed 2405(which do not occur in the equation) to zero needs to make the 2406symmetry to zero. In your ansatz this is not 2407the case because the list of substitutions: 2408{g=0,f=0} 2409leaves this right hand side non-zero: 2410eta_v= - v 2411 4x 2412To fix your ansatz you could, for example, simply multiply all 2413non-vanishing parts on all right hand sides in your ansatz with one 2414and the same unknown constant, say cc, and add cc to the list of unknowns 2415to be computed and to the list of non-vanishing expressions. 2416=============================================================== 2417 2418-------------------------------------------------------------------------- 2419 2420This is LIEPDE - a program for calculating infinitesimal symmetries 2421of single differential equations or systems of de's 2422 2423The ODE under investigation is : 2424 2425 n 2 2426y =x *y 2427 2x 2428 2429for the function(s) : 2430 2431y(x) 2432 2433 2434Start of splitting equations 24354 equations result 2436 2437~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2438This is CRACK - a solver for overdetermined partial differential equations 2439 2440new function: c_173(x) 2441new function: c_174(x) 2442Integration of e_1 2443 2444Substitution of xi_x(x,y) by an expression in 2445c_174(x), c_173(x) found in e_1 : 2446xi_x= - (c_173 + c_174*y) 2447 2448 2449new function: c_175(x) 2450new function: c_176(x) 2451Integration of e_2 2452 2453Substitution of eta_y(x,y) by an expression in 2454c_174(x), c_176(x), c_175(x) found in e_2 : 2455 2 2456eta_y= - (c_174 *y + c_175 + c_176*y) 2457 x 2458 2459 2460Separation of e_3 yields (e_6 e_5 e_7) 2461 2462Separation of e_4 yields (e_11 e_10 e_8 e_9) 2463 2464Substitution of c_174(x) found in e_5 : 2465c_174=0 2466 2467 2468new constant: c_177 2469new constant: c_178 2470Integration of e_11 2471 2472Substitution of c_175(x) by an expression in 2473c_178, c_177 found in e_11 : 2474c_175= - (c_177 + c_178*x) 2475 2476 2477new constant: c_179 2478Integration of e_7 2479 2480Substitution of c_176(x) by an expression in 2481c_173(x), c_179 found in e_7 : 2482 c_173 + c_179 2483 x 2484c_176=---------------- 2485 2 2486 2487 2488Inhomogeneous part: 2489 n n 24904*x *c_178*x + 4*x *c_177 2491 2492can not be integrated explicitly wrt. x 2493The integrated equation: 2494 (2*n)/5 24955*x *c_173 2496 2497 2498An integrating factor has been found: 2499 (2*n)/5 2500 x 2501---------- 2502 x 2503 2504 2505Inhomogeneous part: 2506 (2*n)/5 2507x *c_179 2508 2509can not be integrated explicitly wrt. x 2510New level : 1. 2511 2512CRACK is now called with assuming 2513 25142*n + 5 2515 2516 to be nonzero. 2517The list of inequalities got extended by: 25182*n + 5 2519 2520 2521New level : 1.1. 2522 2523CRACK is now called with assuming 2524 2525 2 2526n + 3*n + 2 2527 2528 to be nonzero. 2529The list of inequalities got extended by: 2530n + 2 2531 2532 2533The list of inequalities got extended by: 2534n + 1 2535 2536 2537Inhomogeneous part: 2538 n n n 2 n 2 2539 4*x *c_177*n*x + 8*x *c_177*x + 4*x *c_178*n*x + 4*x *c_178*x 2540----------------------------------------------------------------- 2541 2 2542 n + 3*n + 2 2543 2544can not be integrated explicitly wrt. x 2545new constant: c_180 2546Inhomogeneous part: 2547 2 n 2 n 2 n 25482*c_180 + 3*c_180*n + c_180*n + 4*x *c_178*x + 4*x *c_178*n*x + 8*x *c_177*x 2549 2550 n 2551 + 4*x *c_177*n*x 2552 2553can not be integrated explicitly wrt. x 2554Not enough integrations to solve for a function, or, 2555substitution prevented through non-linearity. 2556new constant: c_181 2557Integration of e_9 2558 2559Substitution of c_173(x) by an expression in 2560n, c_179, c_181 found in e_9 : 2561 (2*n)/5 2562 - x *c_179*x + 2*c_181*n + 5*c_181 2563c_173=------------------------------------------- 2564 (2*n)/5 2565 x *(2*n + 5) 2566 2567 2568New level : 1.1.1. 2569 2570CRACK is now called with assuming 2571 2572n + 3 2573 2574 to be nonzero. 2575The list of inequalities got extended by: 2576n + 3 2577 2578 2579The list of inequalities got extended by: 2580 3 2 2581n + 6*n + 11*n + 6 2582 2583 2584Separation of e_12 leads to one or more case distinctions. 2585 2586New level : 1.1.1.1. 2587 2588CRACK is now called with assuming 2589 25907*n + 15 2591 2592 to be nonzero. 2593The list of inequalities got extended by: 25947*n + 15 2595 2596 2597Separation of e_12 leads to one or more case distinctions. 2598 2599New level : 1.1.1.1.1. 2600 2601CRACK is now called with assuming 2602 26037*n + 20 2604 2605 to be nonzero. 2606The list of inequalities got extended by: 26077*n + 20 2608 2609 2610Separation of e_12 yields (e_15 e_14 e_13) 2611 2612Substitution of c_177 found in e_15 : 2613c_177=0 2614 2615 2616Substitution of c_178 found in e_14 : 2617c_178=0 2618 2619 2620New level : 1.1.1.1.1.1. 2621 2622CRACK is now called with assuming 2623 2624n + 5 2625 2626 to be nonzero. 2627The list of inequalities got extended by: 2628n + 5 2629 2630 2631New level : 1.1.1.1.1.1.1. 2632 2633CRACK is now called with assuming 2634 2635n 2636 2637 to be nonzero. 2638The list of inequalities got extended by: 2639n 2640 2641The lexicographical ordering of unknowns is changed 2642because n has to be non-zero, giving n a low priority. 2643Old ordering: n,c_179,c_181 2644New ordering: c_179,c_181,n 2645 2646Substitution of c_181 found in e_13 : 2647c_181=0 2648 2649 2650 2651>>>>>>>>> Solution of level 1111111 : 2652Back to level : 1.1.1.1.1.1. 2653 2654New level : 1.1.1.1.1.1.2. 2655 2656CRACK is now called with the assumption 0 = e_21 : 26570=n 2658 2659e_13 is a consequence of e_21 2660 2661Substitution of n found in e_21 : 2662n=0 2663 2664 2665 2666>>>>>>>>> Solution of level 1111112 : 2667Back to level : 1.1.1.1.1.1. 2668 2669This completes the investigation of all cases of a case-distinction. 2670 2671Back to level : 1.1.1.1.1. 2672 2673New level : 1.1.1.1.1.2. 2674 2675 2676The list of inequalities got extended by: 2677n 2678 2679The lexicographical ordering of unknowns is changed 2680because n has to be non-zero, giving n a low priority. 2681Old ordering: n,c_179,c_181 2682New ordering: c_179,c_181,n 2683CRACK is now called with the assumption 0 = e_19 : 26840=n + 5 2685 2686e_13 is a consequence of e_19 2687 2688Substitution of n found in e_19 : 2689n= - 5 2690 2691 2692 2693>>>>>>>>> Solution of level 111112 : 2694Back to level : 1.1.1.1.1. 2695 2696This completes the investigation of all cases of a case-distinction. 2697 2698Back to level : 1.1.1.1. 2699 2700New level : 1.1.1.1.2. 2701 2702 2703The list of inequalities got extended by: 2704n 2705 2706The lexicographical ordering of unknowns is changed 2707because n has to be non-zero, giving n a low priority. 2708Old ordering: n,c_178,c_177,c_179,c_181 2709New ordering: c_178,c_177,c_179,c_181,n 2710CRACK is now called with the assumption 0 = e_13 : 27110=7*n + 20 2712 2713 2714The list of inequalities got extended by: 2715n 2716 2717 2718Substitution of n found in e_13 : 2719 - 20 2720n=------- 2721 7 2722 2723 2724Separation of e_14 yields (e_16 e_15) 2725 2726Substitution of c_177 found in e_16 : 2727c_177=0 2728 2729 2730Substitution of c_181 by an expression in 2731c_178 found in e_15 : 2732 343*c_178 2733c_181=----------- 2734 12 2735 2736 2737 2738>>>>>>>>> Solution of level 11112 : 2739Back to level : 1.1.1.1. 2740 2741This completes the investigation of all cases of a case-distinction. 2742 2743Back to level : 1.1.1. 2744 2745New level : 1.1.1.2. 2746 2747 2748The list of inequalities got extended by: 2749n 2750 2751The lexicographical ordering of unknowns is changed 2752because n has to be non-zero, giving n a low priority. 2753Old ordering: n,c_178,c_177,c_179,c_181 2754New ordering: c_178,c_177,c_179,c_181,n 2755CRACK is now called with the assumption 0 = e_13 : 27560=7*n + 15 2757 2758 2759The list of inequalities got extended by: 2760n 2761 2762 2763Substitution of n found in e_13 : 2764 - 15 2765n=------- 2766 7 2767 2768 2769Separation of e_14 yields (e_15 e_16) 2770 2771Substitution of c_178 found in e_15 : 2772c_178=0 2773 2774 2775Substitution of c_181 by an expression in 2776c_177 found in e_16 : 2777 - 343*c_177 2778c_181=-------------- 2779 12 2780 2781 2782 2783>>>>>>>>> Solution of level 1112 : 2784Back to level : 1.1.1. 2785 2786This completes the investigation of all cases of a case-distinction. 2787 2788Back to level : 1.1. 2789 2790New level : 1.1.2. 2791 2792 2793The list of inequalities got extended by: 2794n 2795 2796The lexicographical ordering of unknowns is changed 2797because n has to be non-zero, giving n a low priority. 2798Old ordering: n,c_178,c_177,c_179,c_181 2799New ordering: c_178,c_177,c_179,c_181,n 2800CRACK is now called with the assumption 0 = e_13 : 28010=n + 3 2802 2803 2804New level : 1.1.2.1. 2805 2806CRACK is now called with assuming 2807 2808n + 3 2809 2810 to be nonzero. 2811The list of inequalities got extended by: 2812n + 3 2813 2814 2815contradiction : 28160=n + 3 2817 2818According to the system of equations, this expression must be zero! 2819 --> Next case. 2820Back to level : 1.1.2. 2821 2822New level : 1.1.2.2. 2823 2824CRACK is now called with the assumption 0 = e_14 : 28250=n + 3 2826 2827 2828e_14 is a consequence of e_13. 2829The list of inequalities got extended by: 2830n 2831 2832 2833Substitution of n found in e_13 : 2834n= - 3 2835 2836 2837Separation of e_15 yields (e_18 e_17 e_16) 2838 2839Substitution of c_177 found in e_18 : 2840c_177=0 2841 2842 2843Substitution of c_178 found in e_17 : 2844c_178=0 2845 2846 2847Substitution of c_181 found in e_16 : 2848c_181=0 2849 2850 2851 2852>>>>>>>>> Solution of level 1122 : 2853Back to level : 1.1.2. 2854 2855This completes the investigation of all cases of a case-distinction. 2856 2857Back to level : 1.1. 2858 2859This completes the investigation of all cases of a case-distinction. 2860 2861Back to level : 1. 2862 2863New level : 1.2. 2864 2865CRACK is now called with the assumption 0 = e_12 : 2866 2 28670=n + 3*n + 2 2868 2869 2870New level : 1.2.1. 2871 2872CRACK is now called with assuming 2873 2874n + 1 2875 2876 to be nonzero. 2877The list of inequalities got extended by: 2878n + 1 2879 2880 2881The list of inequalities got extended by: 2882n 2883 2884The lexicographical ordering of unknowns is changed 2885because n has to be non-zero, giving n a low priority. 2886Old ordering: c_173,n,c_178,c_177,c_179 2887New ordering: c_173,c_178,c_177,c_179,n 2888 2889Substitution of n found in e_12 : 2890n= - 2 2891 2892 2893new constant: c_182 2894Integration of e_16 2895 2896Substitution of c_173(x) by an expression in 2897c_179, c_182 found in e_16 : 2898 4/5 2899c_173=x *c_182 - c_179*x 2900 2901 2902Separation of e_15 yields (e_19 e_18 e_17) 2903 2904Substitution of c_182 found in e_19 : 2905c_182=0 2906 2907 2908Substitution of c_177 found in e_18 : 2909c_177=0 2910 2911 2912Substitution of c_178 found in e_17 : 2913c_178=0 2914 2915 2916 2917>>>>>>>>> Solution of level 121 : 2918Back to level : 1.2. 2919 2920New level : 1.2.2. 2921 2922 2923The list of inequalities got extended by: 2924n 2925 2926The lexicographical ordering of unknowns is changed 2927because n has to be non-zero, giving n a low priority. 2928Old ordering: c_173,n,c_178,c_177,c_179 2929New ordering: c_173,c_178,c_177,c_179,n 2930CRACK is now called with the assumption 0 = e_15 : 29310=n + 1 2932 2933e_12 is a consequence of e_15 2934 2935Substitution of n found in e_15 : 2936n= - 1 2937 2938 2939new constant: c_183 2940Integration of e_17 2941 2942Substitution of c_173(x) by an expression in 2943c_179, c_183 found in e_17 : 2944 2/5 2945 3*x *c_183 - c_179*x 2946c_173=------------------------ 2947 3 2948 2949 2950Separation of e_16 yields (e_21 e_19 e_18) 2951 2952Substitution of c_183 found in e_21 : 2953c_183=0 2954 2955 2956Substitution of c_177 found in e_19 : 2957c_177=0 2958 2959 2960Substitution of c_178 found in e_18 : 2961c_178=0 2962 2963 2964 2965>>>>>>>>> Solution of level 122 : 2966Back to level : 1.2. 2967 2968This completes the investigation of all cases of a case-distinction. 2969 2970Back to level : 1. 2971 2972This completes the investigation of all cases of a case-distinction. 2973 2974New level : 2. 2975 2976 2977The list of inequalities got extended by: 2978n 2979 2980The lexicographical ordering of unknowns is changed 2981because n has to be non-zero, giving n a low priority. 2982Old ordering: c_173,n,c_178,c_177,c_179 2983New ordering: c_173,c_178,c_177,c_179,n 2984CRACK is now called with the assumption 0 = e_12 : 29850=2*n + 5 2986 2987 2988New level : 2.1. 2989 2990CRACK is now called with assuming 2991 2992 2 2993n + 3*n + 2 2994 2995 to be nonzero. 2996The list of inequalities got extended by: 2997n + 2 2998 2999 3000The list of inequalities got extended by: 3001n + 1 3002 3003 3004Substitution of n found in e_12 : 3005 - 5 3006n=------ 3007 2 3008 3009 3010new constant: c_184 3011Integration of e_14 3012 3013Substitution of c_173(x) by an expression in 3014c_179, c_184 found in e_14 : 3015 x*( - log(x)*c_179 + 5*c_184) 3016c_173=------------------------------- 3017 5 3018 3019 3020Separation of e_15 yields (e_18 e_17 e_16) 3021 3022Substitution of c_177 found in e_18 : 3023c_177=0 3024 3025 3026Substitution of c_178 found in e_17 : 3027c_178=0 3028 3029 3030Substitution of c_179 found in e_16 : 3031c_179=0 3032 3033 3034 3035>>>>>>>>> Solution of level 21 : 3036Back to level : 2. 3037 3038New level : 2.2. 3039 3040CRACK is now called with the assumption 0 = e_13 : 3041 2 30420=n + 3*n + 2 3043 3044 3045Substitution of n found in e_12 : 3046 - 5 3047n=------ 3048 2 3049 3050 3051leads to a contradiction in : 3052 2 30530=n + 3*n + 2 3054 3055 3056Back to level : 2. 3057 3058This completes the investigation of all cases of a case-distinction. 3059 3060This completes the investigation of all cases of a case-distinction. 3061 3062This is the end of the CRACK run 3063~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3064 3065-------- 1. Symmetry: 3066 3067 x 3068xi_x=--------- 3069 2*n + 5 3070 3071 - n*y - 2*y 3072eta_y=-------------- 3073 2*n + 5 3074 3075 - y *n - 3*y 3076 x x 3077eta_yx=---------------- 3078 2*n + 5 3079 3080 n 2 n 2 3081 - x *n*y - 4*x *y 3082eta_yxx=---------------------- 3083 2*n + 5 3084 3085 3086-------- 2. Symmetry: 3087 3088n=-2 3089 3090xi_x=x 3091 3092eta_y=0 3093 3094 1 3 3095 - ---*y *n - ---*y 3096 2 x 2 x 3097eta_yx=---------------------- 3098 5 3099 n + --- 3100 2 3101 3102 1 n 2 n 2 3103 - ---*x *n*y - 2*x *y 3104 2 3105eta_yxx=-------------------------- 3106 5 3107 n + --- 3108 2 3109 3110 3111-------- 3. Symmetry: 3112 3113n=-1 3114 3115xi_x=x 3116 3117eta_y= - y 3118 3119 1 3 3120 - ---*y *n - ---*y 3121 2 x 2 x 3122eta_yx=---------------------- 3123 5 3124 n + --- 3125 2 3126 3127 1 n 2 n 2 3128 - ---*x *n*y - 2*x *y 3129 2 3130eta_yxx=-------------------------- 3131 5 3132 n + --- 3133 2 3134 3135 3136-------- 4. Symmetry: 3137 3138 15 3139n= - ---- 3140 7 3141 3142xi_x=7*x 3143 3144eta_y=y 3145 3146 1 3 3147 - ---*y *n - ---*y 3148 2 x 2 x 3149eta_yx=---------------------- 3150 5 3151 n + --- 3152 2 3153 3154 1 n 2 n 2 3155 - ---*x *n*y - 2*x *y 3156 2 3157eta_yxx=-------------------------- 3158 5 3159 n + --- 3160 2 3161 3162-------- 5. Symmetry: 3163 3164 15 3165n= - ---- 3166 7 3167 3168 6/7 3169xi_x=343*x 3170 3171 6/7 3172 147*x *y + 12*x 3173eta_y=------------------- 3174 x 3175 3176eta_yx=0 3177 3178eta_yxx=0 3179 3180 3181-------- 6. Symmetry: 3182 3183n=-5 3184 3185 2 3186xi_x=x 3187 3188eta_y=x*y 3189 3190eta_yx=0 3191 3192eta_yxx=0 3193 3194-------- 7. Symmetry: 3195 3196n=-5 3197 3198xi_x=x 3199 3200eta_y=3*y 3201 3202 1 3 3203 - ---*y *n - ---*y 3204 2 x 2 x 3205eta_yx=---------------------- 3206 5 3207 n + --- 3208 2 3209 3210 1 n 2 n 2 3211 - ---*x *n*y - 2*x *y 3212 2 3213eta_yxx=-------------------------- 3214 5 3215 n + --- 3216 2 3217 3218 3219-------- 8. Symmetry: 3220 3221n=0 3222 3223xi_x=1 3224 3225eta_y=0 3226 3227eta_yx=0 3228 3229eta_yxx=0 3230 3231-------- 9. Symmetry: 3232 3233n=0 3234 3235xi_x=x 3236 3237eta_y= - 2*y 3238 3239 1 3 3240 - ---*y *n - ---*y 3241 2 x 2 x 3242eta_yx=---------------------- 3243 5 3244 n + --- 3245 2 3246 3247 1 n 2 n 2 3248 - ---*x *n*y - 2*x *y 3249 2 3250eta_yxx=-------------------------- 3251 5 3252 n + --- 3253 2 3254 3255 3256-------- 10. Symmetry: 3257 3258 20 3259n= - ---- 3260 7 3261 3262xi_x=7*x 3263 3264eta_y=6*y 3265 3266 1 3 3267 - ---*y *n - ---*y 3268 2 x 2 x 3269eta_yx=---------------------- 3270 5 3271 n + --- 3272 2 3273 3274 1 n 2 n 2 3275 - ---*x *n*y - 2*x *y 3276 2 3277eta_yxx=-------------------------- 3278 5 3279 n + --- 3280 2 3281 3282-------- 11. Symmetry: 3283 3284 20 3285n= - ---- 3286 7 3287 3288 8/7 3289xi_x=343*x 3290 3291 8/7 2 3292 196*x *y - 12*x 3293eta_y=-------------------- 3294 x 3295 3296eta_yx=0 3297 3298eta_yxx=0 3299 3300 3301-------- 12. Symmetry: 3302 3303n=-3 3304 3305xi_x=x 3306 3307eta_y=y 3308 3309 1 3 3310 - ---*y *n - ---*y 3311 2 x 2 x 3312eta_yx=---------------------- 3313 5 3314 n + --- 3315 2 3316 3317 1 n 2 n 2 3318 - ---*x *n*y - 2*x *y 3319 2 3320eta_yxx=-------------------------- 3321 5 3322 n + --- 3323 2 3324 3325 3326-------- 13. Symmetry: 3327 3328 5 3329n= - --- 3330 2 3331 3332xi_x=2*x 3333 3334eta_y=y 3335 3336 1 3 3337 - ---*y *c_179*n - ---*y *c_179 3338 2 x 2 x 3339eta_yx=---------------------------------- 3340 5 3341 n + --- 3342 2 3343 3344 1 n 2 n 2 3345 - ---*x *c_179*n*y - 2*x *c_179*y 3346 2 3347eta_yxx=-------------------------------------- 3348 5 3349 n + --- 3350 2 3351 3352 3353-------- 3354 3355