Problems 501 to 600

Table 2.13: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

501	$5 x y^{''} + (30 + 3 x) y^{'} + 3 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.687

502	$x y^{''} - (x + 4) y^{'} + 3 y = 0$	[_Laguerre]	✓	0.802

503	$2 x y^{''} - (6 + 2 x) y^{'} + y = 0$	[_Laguerre]	✓	1.433

504	$x^{2} y^{''} + (3 x^{2} + 2 x) y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.762

505	$x (1 - x) y^{''} - 3 y^{'} + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.662

506	$x y^{''} + y^{'} - x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.487

507	$x^{2} y^{''} - y^{'} x + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.605

508	$x^{2} y^{''} + (x^{2} - 3 x) y^{'} + 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.687

509	$x^{2} y^{''} + x^{2} y^{'} - 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.798

510	$x^{2} y^{''} + (2 x^{2} - 3 x) y^{'} + 3 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.349

511	$x^{2} y^{''} + x (x + 1) y^{'} - 4 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.774

512	$x^{2} y^{''} + y^{'} x + (x^{2} - \frac{9}{4}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.676

513	$x^{2} y^{''} - x (x + 1) y^{'} + y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.664

514	$x^{2} y^{''} - y^{'} x + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.586

515	$x y^{''} + 3 y^{'} + x y = 0$	[_Lienard]	✓	0.508

516	$x y^{''} - y^{'} + 36 x^{3} y = 0$	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	0.809

517	$x^{2} y^{''} - 5 y^{'} x + (8 + x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.374

518	$36 x^{2} y^{''} + 60 y^{'} x + (9 x^{3} - 5) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.384

519	$16 x^{2} y^{''} + 24 y^{'} x + (144 x^{3} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.463

520	$x^{2} y^{''} + 3 y^{'} x + (x^{2} + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.562

521	$4 x^{2} y^{''} - 12 y^{'} x + (15 + 16 x) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.344

522	$16 x^{2} y^{''} - (- 144 x^{3} + 5) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.546

523	$2 x^{2} y^{''} - 3 y^{'} x - 2 (- x^{5} + 14) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.401

524	$y^{''} + y x^{4} = 0$	[[_Emden, _Fowler]]	✓	0.455

525	$x y^{''} + 4 x^{3} y = 0$	[[_Emden, _Fowler]]	✓	0.446

526	$x y^{''} + 2 y^{'} + x y = 0$	[_Lienard]	✓	0.574

527	$y^{'} = x^{2} + y^{2}$	[[_Riccati, _special]]	✓	1.383

528	$y^{'} = x^{2} + y^{2}$ i.c.	[[_Riccati, _special]]	✓	2.396

529	$y^{'} = x^{2} + y^{2}$ i.c.	[[_Riccati, _special]]	✗	2.017

530	$x^{''} + 4 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.340

531	$x^{''} + 9 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.414

532	$x^{''} - x^{'} - 2 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.332

533	$x^{''} + 8 x^{'} + 15 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.346

534	$x^{''} + x = \sin (2 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.462

535	$x^{''} + 4 x = \cos (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.488

536	$x^{''} + x = \cos (3 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.511

537	$x^{''} + 9 x = 1$ i.c.	[[_2nd_order, _missing_x]]	✓	0.350

538	$x^{''} + 4 x^{'} + 3 x = 1$ i.c.	[[_2nd_order, _missing_x]]	✓	0.342

539	$x^{''} + 3 x^{'} + 2 x = t$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.331

540	$[\begin{matrix} x^{'} = 2 x + y \\ y^{'} = 6 x + 3 y \end{matrix}]$ i.c.	system_of_ODEs	✓	0.454

541	$x^{''} + 6 x^{'} + 25 x = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.494

542	$x^{''} - 6 x^{'} + 8 x = 2$ i.c.	[[_2nd_order, _missing_x]]	✓	0.323

543	$x^{''} - 4 x = 3 t$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.329

544	$x^{''} + 4 x^{'} + 8 x = e^{- t}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.498

545	$x^{'''} + x^{''} - 6 x^{'} = 0$ i.c.	[[_3rd_order, _missing_x]]	✓	0.347

546	$x^{''''} - x = 0$ i.c.	[[_high_order, _missing_x]]	✓	0.427

547	$x^{''''} + x = 0$ i.c.	[[_high_order, _missing_x]]	✓	0.609

548	$x^{''''} + 13 x^{''} + 36 x = 0$ i.c.	[[_high_order, _missing_x]]	✓	0.549

549	$x^{''''} + 8 x^{''} + 16 x = 0$ i.c.	[[_high_order, _missing_x]]	✓	0.498

550	$x^{''''} + 2 x^{''} + x = e^{2 t}$ i.c.	[[_high_order, _with_linear_symmetries]]	✓	0.549

551	$x^{''} + 4 x^{'} + 13 x = t e^{- t}$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.541

552	$x^{''} + 6 x^{'} + 18 x = \cos (2 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.557

553	$x^{''} + 9 x = 6 \cos (3 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.386

554	$x^{''} + \frac{2 x^{'}}{5} + \frac{226 x}{25} = 6 e^{- \frac{t}{5}} \cos (3 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.495

555	$t x^{''} + (t - 2) x^{'} + x = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.319

556	$t x^{''} + (3 t - 1) x^{'} + 3 x = 0$ i.c.	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.323

557	$t x^{''} - (4 t + 1) x^{'} + 2 (2 t + 1) x = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.351

558	$t x^{''} + 2 (t - 1) x^{'} - 2 x = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.361

559	$t x^{''} - 2 x^{'} + t x = 0$ i.c.	[_Lienard]	✓	0.347

560	$t x^{''} + (4 t - 2) x^{'} + (13 t - 4) x = 0$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.327

561	$x^{''} + 4 x = f (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.894

562	$x^{''} + 2 x^{'} + x = f (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.494

563	$x^{''} + 4 x^{'} + 13 x = f (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	2.149

564	$x^{''} + 4 x = δ (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.285

565	$x^{''} + 4 x = δ (t) + δ (t - π)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.789

566	$x^{''} + 4 x^{'} + 4 x = 1 + δ (t - 2)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.761

567	$x^{''} + 2 x^{'} + x = t + δ (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.285

568	$x^{''} + 2 x^{'} + 2 x = 2 δ (t - π)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.774

569	$x^{''} + 9 x = δ (t - 3 π) + \cos (3 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.738

570	$x^{''} + 4 x^{'} + 5 x = δ (t - π) + δ (t - 2 π)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.290

571	$x^{''} + 2 x^{'} + x = δ (t) - δ (t - 2)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.819

572	$x^{''} + 4 x = f (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.877

573	$x^{''} + 6 x^{'} + 9 x = f (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.463

574	$x^{''} + 6 x^{'} + 8 x = f (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.487

575	$x^{''} + 4 x^{'} + 8 x = f (t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.943

576	$[\begin{matrix} x^{'} = y \\ y^{'} = - x \end{matrix}]$	system_of_ODEs	✓	0.397

577	$[\begin{matrix} x^{'} = y \\ y^{'} = x \end{matrix}]$	system_of_ODEs	✓	0.312

578	$[\begin{matrix} x^{'} = - 2 y \\ y^{'} = 2 x \end{matrix}]$ i.c.	system_of_ODEs	✓	0.452

579	$[\begin{matrix} x^{'} = 10 y \\ y^{'} = - 10 x \end{matrix}]$ i.c.	system_of_ODEs	✓	0.485

580	$[\begin{matrix} x^{'} = \frac{y}{2} \\ y^{'} = - 8 x \end{matrix}]$	system_of_ODEs	✓	0.451

581	$[\begin{matrix} x^{'} = 8 y \\ y^{'} = - 2 x \end{matrix}]$	system_of_ODEs	✓	0.410

582	$[\begin{matrix} x^{'} = y \\ y^{'} = 6 x - y \end{matrix}]$ i.c.	system_of_ODEs	✓	0.442

583	$[\begin{matrix} x^{'} = - y \\ y^{'} = 10 x - 7 y \end{matrix}]$ i.c.	system_of_ODEs	✓	0.478

584	$[\begin{matrix} x^{'} = - y \\ y^{'} = 13 x + 4 y \end{matrix}]$ i.c.	system_of_ODEs	✓	0.648

585	$[\begin{matrix} x^{'} = y \\ y^{'} = - 9 x + 6 y \end{matrix}]$	system_of_ODEs	✓	0.349

586	$[\begin{matrix} 10 x_{1}^{'} = - x_{1} + x_{3} \\ 10 x_{2}^{'} = x_{1} - x_{2} \\ 10 x_{3}^{'} = x_{2} - x_{3} \end{matrix}]$	system_of_ODEs	✓	1.047

587	$[\begin{matrix} x^{'} = - x + 3 y \\ y^{'} = 2 y \end{matrix}]$	system_of_ODEs	✓	0.348

588	$[\begin{matrix} x^{'} = x - 2 y \\ y^{'} = 2 x - 3 y \end{matrix}]$	system_of_ODEs	✓	0.307

589	$[\begin{matrix} x^{'} = - 3 x + 2 y \\ y^{'} = - 3 x + 4 y \end{matrix}]$ i.c.	system_of_ODEs	✓	0.507

590	$[\begin{matrix} x^{'} = 3 x - y \\ y^{'} = 5 x - 3 y \end{matrix}]$ i.c.	system_of_ODEs	✓	0.520

591	$[\begin{matrix} x^{'} = - 3 x - 4 y \\ y^{'} = 2 x + y \end{matrix}]$	system_of_ODEs	✓	0.569

592	$[\begin{matrix} x^{'} = x + 9 y \\ y^{'} = - 2 x - 5 y \end{matrix}]$ i.c.	system_of_ODEs	✓	0.574

593	$[\begin{matrix} x^{'} = 4 x + y + 2 t \\ y^{'} = - 2 x + y \end{matrix}]$	system_of_ODEs	✓	0.562

594	$[\begin{matrix} x^{'} = 2 x + y \\ y^{'} = x + 2 y - e^{2 t} \end{matrix}]$	system_of_ODEs	✓	0.572

595	$[\begin{matrix} x^{'} = 2 x - 3 y + 2 \sin (2 t) \\ y^{'} = x - 2 y - \cos (2 t) \end{matrix}]$	system_of_ODEs	✓	1.026

596	$[\begin{matrix} x^{'} + 2 y^{'} = 4 x + 5 y \\ 2 x^{'} - y^{'} = 3 x \end{matrix}]$ i.c.	system_of_ODEs	✓	0.481

597	$[\begin{matrix} - x^{'} + 2 y^{'} = x + 3 y + e^{t} \\ 3 x^{'} - 4 y^{'} = x - 15 y + e^{- t} \end{matrix}]$	system_of_ODEs	✓	0.987

598	$[\begin{matrix} x^{'} = x + 2 y + z \\ y^{'} = 6 x - y \\ z^{'} = - x - 2 y - z \end{matrix}]$	system_of_ODEs	✓	0.639

599	$[\begin{matrix} x^{'} = x - 2 y \\ y^{'} = - 4 x + 4 y - 2 z \\ z^{'} = - 4 y + 4 z \end{matrix}]$	system_of_ODEs	✓	84.663

600	$[\begin{matrix} x^{'} = y + z + e^{- t} \\ y^{'} = x + z \\ z^{'} = x + y \end{matrix}]$	system_of_ODEs	✓	0.643

2.2.6 Problems 501 to 600