Problems 401 to 500

Table 2.11: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

401	$y^{'} = 4 y$	[_quadrature]	✓	0.264

402	$2 y^{'} + 3 y = 0$	[_quadrature]	✓	0.273

403	$y^{'} + 2 x y = 0$	[_separable]	✓	0.333

404	$y^{'} = x y$	[_separable]	✓	0.364

405	$(x - 2) y^{'} + y = 0$	[_separable]	✓	0.319

406	$(2 x - 1) y^{'} + 2 y = 0$	[_separable]	✓	0.334

407	$2 (x + 1) y^{'} + y = 0$	[_separable]	✓	0.338

408	$(x - 1) y^{'} + 2 y = 0$	[_separable]	✓	0.385

409	$2 (x - 1) y^{'} = 3 y$	[_separable]	✓	0.347

410	$y^{''} = y$	[[_2nd_order, _missing_x]]	✓	0.364

411	$y^{''} = 4 y$	[[_2nd_order, _missing_x]]	✓	0.342

412	$y^{''} + 9 y = 0$	[[_2nd_order, _missing_x]]	✓	0.385

413	$y^{''} + y = x$	[[_2nd_order, _with_linear_symmetries]]	✓	0.338

414	$y^{'} x + y = 0$	[_separable]	✓	0.230

415	$2 y^{'} x = y$	[_separable]	✓	0.269

416	$x^{2} y^{'} + y = 0$	[_separable]	✗	0.118

417	$x^{3} y^{'} = 2 y$	[_separable]	✗	0.132

418	$y^{''} + 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.395

419	$y^{''} - 4 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.377

420	$y^{''} - 2 y^{'} + y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.474

421	$y^{''} + y^{'} - 2 y = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.482

422	$x^{2} y^{''} + x^{2} y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.786

423	$y^{''} = y^{'} + y$ i.c.	[[_2nd_order, _missing_x]]	✓	0.471

424	$y^{'} = 1 + y^{2}$ i.c.	[_quadrature]	✓	0.246

425	$y^{'''} = y$ i.c.	[[_3rd_order, _missing_x]]	✓	0.161

426	$(x^{2} - 1) y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.459

427	$(x^{2} + 2) y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.476

428	$y^{''} + y^{'} x + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.363

429	$(x^{2} + 1) y^{''} + 6 y^{'} x + 4 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.489

430	$(x^{2} - 3) y^{''} + 2 y^{'} x = 0$	[[_2nd_order, _missing_y]]	✓	0.384

431	$(x^{2} - 1) y^{''} - 6 y^{'} x + 12 y = 0$	[_Gegenbauer]	✓	0.411

432	$(x^{2} + 3) y^{''} - 7 y^{'} x + 16 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.477

433	$(- x^{2} + 2) y^{''} - y^{'} x + 16 y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.484

434	$(x^{2} - 1) y^{''} + 8 y^{'} x + 12 y = 0$	[_Gegenbauer]	✓	0.508

435	$3 y^{''} + y^{'} x - 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.418

436	$5 y^{''} - 2 y^{'} x + 10 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.402

437	$y^{''} - x^{2} y^{'} - 3 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.382

438	$y^{''} + x^{2} y^{'} + 2 x y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.414

439	$y^{''} + x y = 0$	[[_Emden, _Fowler]]	✓	0.335

440	$y^{''} + x^{2} y = 0$	[[_Emden, _Fowler]]	✓	0.277

441	$(x^{2} + 1) y^{''} + 2 y^{'} x - 2 y = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.459

442	$y^{''} + y^{'} x - 2 y = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.391

443	$y^{''} + (x - 1) y^{'} + y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.399

444	$(- x^{2} + 2 x) y^{''} - 6 (x - 1) y^{'} - 4 y = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.567

445	$(x^{2} - 6 x + 10) y^{''} - 4 (x - 3) y^{'} + 6 y = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.458

446	$(4 x^{2} + 16 x + 17) y^{''} = 8 y$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.456

447	$(x^{2} + 6 x) y^{''} + (3 x + 9) y^{'} - 3 y = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.482

448	$y^{''} + (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.359

449	$(x^{2} - 1) y^{''} + 2 y^{'} x + 2 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.556

450	$y^{''} + x^{2} y^{'} + x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.435

451	$(x^{3} + 1) y^{''} + y x^{4} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.437

452	$y^{''} + y^{'} x + (2 x^{2} + 1) y = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.484

453	$y^{''} + e^{- x} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.571

454	$\cos (x) y^{''} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.791

455	$x y^{''} + y^{'} \sin (x) + x y = 0$	[_Lienard]	✓	1.558

456	$y^{''} - 2 y^{'} x + 2 α y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.457

457	$x y^{''} + (- x^{3} + x) y^{'} + y \sin (x) = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.883

458	$x y^{''} + x^{2} y^{'} + (e^{x} - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.762

459	$x^{2} y^{''} + \cos (x) y^{'} + x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.257

460	$3 x^{3} y^{''} + 2 x^{2} y^{'} + (- x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.168

461	$x (x + 1) y^{''} + 2 y^{'} + 3 x y = 0$	[[_Emden, _Fowler]]	✓	1.477

462	$x^{2} (- x^{2} + 1) y^{''} + 2 y^{'} x - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.646

463	$x^{2} y^{''} + 6 y^{'} \sin (x) + 6 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.758

464	$(2 x^{3} + 6 x^{2}) y^{''} + 21 y^{'} x + 9 (x^{2} - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.933

465	$(1 - x) y^{''} + y^{'} x + x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.417

466	${(1 - x)}^{2} y^{''} + (2 x - 2) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.556

467	$(- x^{2} + 1) y^{''} - 2 y^{'} x + 12 y = 0$	[_Gegenbauer]	✓	0.468

468	${(x - 2)}^{3} y^{''} + 3 {(x - 2)}^{2} y^{'} + x^{3} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.580

469	$(x^{2} - 4) y^{''} + (x - 2) y^{'} + (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.524

470	${(x^{2} - 9)}^{2} y^{''} + (x^{2} + 9) y^{'} + (x^{2} + 4) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.700

471	${(x - 2)}^{2} y^{''} - (x^{2} - 4) y^{'} + (x + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.546

472	$x^{3} (1 - x) y^{''} + (3 x + 2) y^{'} + x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	0.184

473	$4 x y^{''} + 2 y^{'} + y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.712

474	$2 x y^{''} + 3 y^{'} - y = 0$	[[_Emden, _Fowler]]	✓	0.699

475	$2 x y^{''} - y^{'} - y = 0$	[[_Emden, _Fowler]]	✓	0.684

476	$3 x y^{''} + 2 y^{'} + 2 y = 0$	[[_Emden, _Fowler]]	✓	0.704

477	$2 x^{2} y^{''} + y^{'} x - (2 x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.617

478	$2 x^{2} y^{''} + y^{'} x - (- 2 x^{2} + 3) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.631

479	$6 x^{2} y^{''} + 7 y^{'} x - (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.637

480	$3 x^{2} y^{''} + 2 y^{'} x + x^{2} y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.536

481	$2 x y^{''} + (x + 1) y^{'} + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.717

482	$2 x y^{''} + (- 2 x^{2} + 1) y^{'} - 4 x y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.605

483	$x y^{''} + 2 y^{'} + 9 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.655

484	$x y^{''} + 2 y^{'} - 4 x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.570

485	$4 x y^{''} + 8 y^{'} + x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.618

486	$x y^{''} - y^{'} + 4 x^{3} y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.537

487	$4 x^{2} y^{''} - 4 y^{'} x + (- 4 x^{2} + 3) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.724

488	$2 x^{2} y^{''} + x (x + 1) y^{'} - (2 x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.754

489	$(5 x^{3} + 2 x^{2}) y^{''} + (- x^{2} + 3 x) y^{'} - (x + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.897

490	$2 x^{2} y^{''} + y^{'} \sin (x) - \cos (x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.740

491	$x^{2} y^{''} + (3 x - 1) y^{'} + y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✗	0.207

492	$x^{3} y^{''} - y^{'} x + y = 0$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✗	0.145

493	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{1}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.658

494	$x^{2} y^{''} + y^{'} x + (x^{2} - 1) y = 0$	[_Bessel]	✓	1.219

495	$x^{2} y^{''} + y^{'} x + (1 - x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.646

496	$x (x - 1) {(x + 1)}^{2} y^{''} + 2 x (x - 3) (x + 1) y^{'} - 2 (x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.597

497	$x (1 - x) y^{''} + (γ - (α + β + 1) x) y^{'} - α β y = 0$	[_Jacobi]	✓	1.110

498	$x y^{''} + (3 - x) y^{'} - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.665

499	$x y^{''} + (5 - x) y^{'} - y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.677

500	$x y^{''} + (3 x + 5) y^{'} + 3 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.760

2.2.5 Problems 401 to 500