Problems 4501 to 4600

Table 2.93: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

4501	$y^{''} - 3 y^{'} + 2 y = \sin (e^{- x})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.788

4502	$y^{''} + y = \sec {(x)}^{3}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.762

4503	$y^{''} - y = \frac{1}{\sqrt{1 - e^{2 x}}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.698

4504	$y^{''} - y = e^{- 2 x} \sin (e^{- x})$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.131

4505	$y^{''} + 2 y^{'} + y = 15 e^{- x} \sqrt{x + 1}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.661

4506	$y^{''} + 4 y = 2 \tan (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.880

4507	$y^{''} - 2 y^{'} + y = \frac{e^{2 x}}{{(e^{x} + 1)}^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.702

4508	$y^{''} + y^{'} = \frac{1}{e^{x} + 1}$	[[_2nd_order, _missing_y]]	✓	1.184

4509	$x^{2} y^{''} - y^{'} x + y = \ln (x)$	[[_2nd_order, _with_linear_symmetries]]	✓	1.590

4510	$x^{2} y^{''} + 3 y^{'} x + 5 y = \frac{5 \ln (x)}{x^{2}}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	22.308

4511	$x^{3} y^{'''} + 2 x^{2} y^{''} - y^{'} x + y = 9 x^{2} \ln (x)$	[[_3rd_order, _exact, _linear, _nonhomogeneous]]	✓	0.266

4512	${(x - 2)}^{2} y^{''} - 3 (x - 2) y^{'} + 4 y = x$	[[_2nd_order, _with_linear_symmetries]]	✓	0.999

4513	$x^{3} y^{'''} + 3 x^{2} y^{''} + y^{'} x - y = x^{2}$	[[_3rd_order, _with_linear_symmetries]]	✓	1.050

4514	$y^{''} + 4 y^{'} + 3 y = 60 \cos (3 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.487

4515	$y^{''} + y^{'} - 2 y = 9 e^{- 2 t}$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.340

4516	$y^{''} - y^{'} - 2 y = 2 t^{2} + 1$ i.c.	[[_2nd_order, _with_linear_symmetries]]	✓	0.333

4517	$y^{''} + 4 y = 8 \sin (2 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.466

4518	$y^{''} - 2 y^{'} + y = 4 e^{- t} + 2 e^{t}$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.377

4519	$y^{''} - 2 y^{'} + 2 y = 8 \sin (t) e^{- t}$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.554

4520	$y^{''} - 2 y^{'} + 5 y = 8 e^{t} \sin (2 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.431

4521	$y^{''} + y^{'} - 2 y = 54 t e^{- 2 t}$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.373

4522	$y^{''} - y^{'} - 2 y = 9 e^{2 t} Heaviside (t - 1)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.308

4523	$y^{''} + 2 y^{'} + y = 2 \sin (t) Heaviside (t - π)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.905

4524	$y^{''} + 4 y = 8 Heaviside (t - π) \sin (2 t)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.841

4525	$y^{''} + 4 y = 8 (t^{2} + t - 1) Heaviside (t - 2)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.496

4526	$y^{''} - 3 y^{'} + 2 y = e^{t} Heaviside (t - 2)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.963

4527	$y^{''} - 5 y^{'} + 6 y = δ (t - 2)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.150

4528	$y^{''} + 4 y = 4 Heaviside (t - π) + 2 δ (t - π)$ i.c.	[[_2nd_order, _linear, _nonhomogeneous]]	✓	1.051

4529	$y^{'''} - y^{''} + 4 y^{'} - 4 y = 10 e^{- t}$ i.c.	[[_3rd_order, _with_linear_symmetries]]	✓	0.562

4530	$y^{''''} - 5 y^{''} + 4 y = 120 e^{3 t} Heaviside (t - 1)$ i.c.	[[_high_order, _linear, _nonhomogeneous]]	✓	5.419

4531	$y^{''''} + 3 y^{''} - 4 y = 40 t^{2} Heaviside (t - 2)$ i.c.	[[_high_order, _linear, _nonhomogeneous]]	✓	6.327

4532	$y^{''''} + 4 y = (2 t^{2} + t + 1) δ (t - 1)$ i.c.	[[_high_order, _linear, _nonhomogeneous]]	✓	2.799

4533	$[\begin{matrix} x^{'} + 2 x - y = 0 \\ x + y^{'} - 2 y = 0 \end{matrix}]$	system_of_ODEs	✓	0.540

4534	$[\begin{matrix} 2 x^{'} + x - 5 y^{'} - 4 y = 0 \\ - y^{'} - 2 x + y = 0 \end{matrix}]$	system_of_ODEs	✓	0.403

4535	$[\begin{matrix} x^{'} - x + 3 y = 0 \\ 3 x - y^{'} + y = 0 \end{matrix}]$	system_of_ODEs	✓	0.473

4536	$[\begin{matrix} x^{''} + x^{'} + y^{'} - 2 y = 0 \\ x^{'} + x - y^{'} = 0 \end{matrix}]$	system_of_ODEs	✓	0.049

4537	$[\begin{matrix} x^{''} - 3 x - 4 y = 0 \\ x + y^{''} + y = 0 \end{matrix}]$	system_of_ODEs	✓	0.049

4538	$[\begin{matrix} y_{1}^{'} - y_{2} = 0 \\ 4 y_{1} + y_{2}^{'} - 4 y_{2} - 2 y_{3} = 0 \\ - 2 y_{1} + y_{2} + y_{3}^{'} + y_{3} = 0 \end{matrix}]$	system_of_ODEs	✓	0.525

4539	$[\begin{matrix} y_{1}^{'} - 2 y_{1} + 3 y_{2} - 3 y_{3} = 0 \\ - 4 y_{1} + y_{2}^{'} + 5 y_{2} - 3 y_{3} = 0 \\ - 4 y_{1} + 4 y_{2} + y_{3}^{'} - 2 y_{3} = 0 \end{matrix}]$	system_of_ODEs	✓	0.550

4540	$[\begin{matrix} x^{'} + x + 2 y = 8 \\ 2 x + y^{'} - 2 y = 2 e^{- t} - 8 \end{matrix}]$	system_of_ODEs	✓	0.599

4541	$[\begin{matrix} x^{'} = 2 x - 3 y + t e^{- t} \\ y^{'} = 2 x - 3 y + e^{- t} \end{matrix}]$	system_of_ODEs	✓	0.510

4542	$[\begin{matrix} x^{'} - x - 2 y = e^{t} \\ - 4 x + y^{'} - 3 y = 1 \end{matrix}]$	system_of_ODEs	✓	0.915

4543	$[\begin{matrix} x^{'} - 4 x + 3 y = \sin (t) \\ - 2 x + y^{'} + y = - 2 \cos (t) \end{matrix}]$	system_of_ODEs	✓	0.743

4544	$[\begin{matrix} x^{'} - y = 0 \\ - x + y^{'} = e^{t} + e^{- t} \end{matrix}]$	system_of_ODEs	✓	0.817

4545	$[\begin{matrix} x^{'} + 2 x + 5 y = 0 \\ - x + y^{'} - 2 y = \sin (2 t) \end{matrix}]$	system_of_ODEs	✓	0.791

4546	$[\begin{matrix} x^{'} - 2 x + 2 y^{'} = - 4 e^{2 t} \\ 2 x^{'} - 3 x + 3 y^{'} - y = 0 \end{matrix}]$	system_of_ODEs	✓	0.560

4547	$[\begin{matrix} 3 x^{'} + 2 x + y^{'} - 6 y = 5 e^{t} \\ 4 x^{'} + 2 x + y^{'} - 8 y = 5 e^{t} + 2 t - 3 \end{matrix}]$	system_of_ODEs	✓	0.856

4548	$[\begin{matrix} x^{'} - 5 x + 3 y = 2 e^{3 t} \\ - x + y^{'} - y = 5 e^{- t} \end{matrix}]$	system_of_ODEs	✓	1.115

4549	$[\begin{matrix} x^{'} - 2 x + y = 0 \\ x + y^{'} - 2 y = - 5 e^{t} \sin (t) \end{matrix}]$	system_of_ODEs	✓	0.770

4550	$[\begin{matrix} x^{'} + 4 x + 2 y = \frac{2}{e^{t} - 1} \\ 6 x - y^{'} + 3 y = \frac{3}{e^{t} - 1} \end{matrix}]$	system_of_ODEs	✓	0.083

4551	$[\begin{matrix} x^{'} - x + y = \sec (t) \\ - 2 x + y^{'} + y = 0 \end{matrix}]$	system_of_ODEs	✓	0.951

4552	$[\begin{matrix} x^{'} - x - 2 y = 16 e^{t} t \\ 2 x - y^{'} - 2 y = 0 \end{matrix}]$ i.c.	system_of_ODEs	✓	0.299

4553	$[\begin{matrix} x^{'} - 2 x + y = 5 e^{t} \cos (t) \\ x + y^{'} - 2 y = 10 e^{t} \sin (t) \end{matrix}]$ i.c.	system_of_ODEs	✓	0.432

4554	$[\begin{matrix} x^{'} - 4 x + 3 y = \sin (t) \\ 2 x + y^{'} - y = 2 \cos (t) \end{matrix}]$ i.c.	system_of_ODEs	✓	0.410

4555	$[\begin{matrix} x^{'} - 2 x - y = 2 e^{t} \\ x - y^{'} + 2 y = 3 e^{4 t} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.263

4556	$[\begin{matrix} x^{''} + x^{'} + y^{'} - 2 y = 40 e^{3 t} \\ x^{'} + x - y^{'} = 36 e^{t} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.038

4557	$[\begin{matrix} x^{'} - 2 x - y = 2 e^{t} \\ y^{'} - 2 y - 4 z = 4 e^{2 t} \\ x - z^{'} - z = 0 \end{matrix}]$ i.c.	system_of_ODEs	✓	0.265

4558	$[\begin{matrix} x^{''} + 2 x - 2 y^{'} = 0 \\ 3 x^{'} + y^{''} - 8 y = 240 e^{t} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.033

4559	$[\begin{matrix} x^{'} - x - 2 y = 0 \\ x - y^{'} = 15 \cos (t) Heaviside (t - π) \end{matrix}]$ i.c.	system_of_ODEs	✓	0.508

4560	$[\begin{matrix} x^{'} - x + y = 2 \sin (t) (1 - Heaviside (t - π)) \\ 2 x - y^{'} - y = 0 \end{matrix}]$ i.c.	system_of_ODEs	✓	0.574

4561	$[\begin{matrix} 2 x^{'} + x - 5 y^{'} - 4 y = 28 e^{t} Heaviside (t - 2) \\ 3 x^{'} - 2 x - 4 y^{'} + y = 0 \end{matrix}]$ i.c.	system_of_ODEs	✓	0.517

4562	$[\begin{matrix} x_{1}^{'} = x_{1} - x_{2} \\ x_{2}^{'} = - 4 x_{1} + x_{2} \end{matrix}]$	system_of_ODEs	✓	0.341

4563	$[\begin{matrix} x_{1}^{'} = x_{1} - 3 x_{2} \\ x_{2}^{'} = 3 x_{1} + x_{2} \end{matrix}]$	system_of_ODEs	✓	0.472

4564	$[\begin{matrix} x_{1}^{'} = 5 x_{1} + 3 x_{2} \\ x_{2}^{'} = - 3 x_{1} - x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.498

4565	$[\begin{matrix} x_{1}^{'} = 2 x_{1} - x_{2} + x_{3} \\ x_{2}^{'} = x_{1} + 2 x_{2} - x_{3} \\ x_{3}^{'} = x_{1} - x_{2} + 2 x_{3} \end{matrix}]$	system_of_ODEs	✓	0.495

4566	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - x_{2} + x_{3} \\ x_{2}^{'} = x_{1} + x_{2} + x_{3} \\ x_{3}^{'} = 4 x_{1} - x_{2} + 4 x_{3} \end{matrix}]$	system_of_ODEs	✓	0.603

4567	$[\begin{matrix} x_{1}^{'} = 2 x_{1} + x_{2} \\ x_{2}^{'} = x_{1} + 3 x_{2} - x_{3} \\ x_{3}^{'} = - x_{1} + 2 x_{2} + 3 x_{3} \end{matrix}]$	system_of_ODEs	✓	0.957

4568	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - 2 x_{2} - x_{3} \\ x_{2}^{'} = 3 x_{1} - 4 x_{2} - 3 x_{3} \\ x_{3}^{'} = 2 x_{1} - 4 x_{2} \end{matrix}]$	system_of_ODEs	✓	0.572

4569	$[\begin{matrix} x_{1}^{'} = x_{1} - x_{2} + x_{3} \\ x_{2}^{'} = x_{1} + x_{2} - x_{3} \\ x_{3}^{'} = - 2 x_{2} + 2 x_{3} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.485

4570	$[\begin{matrix} x_{1}^{'} = - x_{1} + x_{2} - 2 x_{3} \\ x_{2}^{'} = 4 x_{1} + x_{2} \\ x_{3}^{'} = 2 x_{1} + x_{2} - x_{3} \end{matrix}]$	system_of_ODEs	✓	0.526

4571	$[\begin{matrix} x_{1}^{'} = 2 x_{1} + x_{2} + 26 \sin (t) \\ x_{2}^{'} = 3 x_{1} + 4 x_{2} \end{matrix}]$	system_of_ODEs	✓	0.829

4572	$[\begin{matrix} x_{1}^{'} = - x_{1} + 8 x_{2} + 9 t \\ x_{2}^{'} = x_{1} + x_{2} + 3 e^{- t} \end{matrix}]$	system_of_ODEs	✓	1.615

4573	$[\begin{matrix} x_{1}^{'} = - x_{1} + 2 x_{2} \\ x_{2}^{'} = - 3 x_{1} + 4 x_{2} + \frac{e^{3 t}}{1 + e^{2 t}} \end{matrix}]$	system_of_ODEs	✓	0.086

4574	$[\begin{matrix} x_{1}^{'} = - 4 x_{1} - 2 x_{2} + \frac{2}{e^{t} - 1} \\ x_{2}^{'} = 6 x_{1} + 3 x_{2} - \frac{3}{e^{t} - 1} \end{matrix}]$	system_of_ODEs	✓	0.083

4575	$[\begin{matrix} x_{1}^{'} = x_{1} + x_{2} + e^{2 t} \\ x_{2}^{'} = - 2 x_{1} + 3 x_{2} \end{matrix}]$	system_of_ODEs	✓	0.805

4576	$[\begin{matrix} x_{1}^{'} = - x_{1} - 5 x_{2} \\ x_{2}^{'} = x_{1} + x_{2} + \frac{4}{\sin (2 t)} \end{matrix}]$	system_of_ODEs	✓	1.302

4577	$[\begin{matrix} x_{1}^{'} = 2 x_{1} + x_{2} + 27 t \\ x_{2}^{'} = - x_{1} + 4 x_{2} \end{matrix}]$	system_of_ODEs	✓	0.496

4578	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - x_{2} + e^{t} \\ x_{2}^{'} = 4 x_{1} - x_{2} \end{matrix}]$	system_of_ODEs	✓	0.498

4579	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - 2 x_{2} \\ x_{2}^{'} = 2 x_{1} - x_{2} + 35 e^{t} t^{3 / 2} \end{matrix}]$	system_of_ODEs	✓	0.556

4580	$[\begin{matrix} x_{1}^{'} = x_{1} - x_{2} + x_{3} \\ x_{2}^{'} = x_{1} + x_{2} - x_{3} + 6 e^{- t} \\ x_{3}^{'} = 2 x_{1} - x_{2} \end{matrix}]$	system_of_ODEs	✓	0.876

4581	$[\begin{matrix} x_{1}^{'} = x_{1} - 2 x_{2} - x_{3} \\ x_{2}^{'} = - x_{1} + x_{2} + x_{3} + 12 t \\ x_{3}^{'} = x_{1} - x_{3} \end{matrix}]$	system_of_ODEs	✓	0.973

4582	$[\begin{matrix} x_{1}^{'} = - 3 x_{1} + 4 x_{2} - 2 x_{3} + e^{t} \\ x_{2}^{'} = x_{1} + x_{2} \\ x_{3}^{'} = 6 x_{1} - 6 x_{2} + 5 x_{3} \end{matrix}]$	system_of_ODEs	✓	162.213

4583	$[\begin{matrix} x_{1}^{'} = x_{1} - x_{2} - x_{3} + 4 e^{t} \\ x_{2}^{'} = x_{1} + x_{2} \\ x_{3}^{'} = 3 x_{1} + x_{3} \end{matrix}]$	system_of_ODEs	✓	1.188

4584	$[\begin{matrix} x_{1}^{'} = 2 x_{1} - x_{2} + 2 x_{3} \\ x_{2}^{'} = x_{1} + 2 x_{3} \\ x_{3}^{'} = - 2 x_{1} + x_{2} - x_{3} + 4 \sin (t) \end{matrix}]$	system_of_ODEs	✓	1.510

4585	$[\begin{matrix} x_{1}^{'} = 4 x_{1} - x_{2} - x_{3} + e^{3 t} \\ x_{2}^{'} = x_{1} + 2 x_{2} - x_{3} \\ x_{3}^{'} = x_{1} + x_{2} + 2 x_{3} \end{matrix}]$	system_of_ODEs	✓	2.497

4586	$[\begin{matrix} x_{1}^{'} = 2 x_{1} - x_{2} - x_{3} + 2 e^{2 t} \\ x_{2}^{'} = 3 x_{1} - 2 x_{2} - 3 x_{3} \\ x_{3}^{'} = - x_{1} + x_{2} + 2 x_{3} \end{matrix}]$	system_of_ODEs	✓	0.681

4587	$[\begin{matrix} x_{1}^{'} = 2 x_{1} - x_{3} + 24 t \\ x_{2}^{'} = x_{1} - x_{2} \\ x_{3}^{'} = 3 x_{1} - x_{2} - x_{3} \end{matrix}]$	system_of_ODEs	✓	0.576

4588	$y^{''} - x y = 0$	[[_Emden, _Fowler]]	✓	0.301

4589	$(x^{2} + 1) y^{''} + 4 y^{'} x + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.471

4590	$y^{''} + x y = 0$	[[_Emden, _Fowler]]	✓	0.303

4591	$(- x^{2} + 1) y^{''} + y = 0$	[_Gegenbauer]	✓	0.429

4592	$y^{''} - 2 x^{2} y = 0$	[[_Emden, _Fowler]]	✓	0.296

4593	$y^{''} - 2 x^{2} y^{'} + x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.413

4594	$(x^{2} - 1) y^{''} + (4 x - 1) y^{'} + 2 y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.541

4595	$y^{''} + (\cos (x) + 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.549

4596	$y^{''} + y^{'} \sin (x) + \cos (x) y = 0$	[[_2nd_order, _exact, _linear, _homogeneous]]	✓	0.755

4597	$x y^{''} + y^{'} - x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.486

4598	$x y^{''} + (1 - x) y^{'} + k y = 0$	[_Laguerre]	✓	0.847

4599	$x^{2} y^{''} + (- 2 x^{2} + x) y^{'} - x y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.663

4600	$x^{2} y^{''} - (x^{2} + 2 x) y^{'} + 2 y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	1.316

2.2.46 Problems 4501 to 4600