Problems 7501 to 7600

Table 2.153: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

7501	$(\cos (x) - \sin (x)) y^{''} - 2 y^{'} \sin (x) + y (\cos (x) + \sin (x)) = {(\cos (x) - \sin (x))}^{2}$	[[_2nd_order, _linear, _nonhomogeneous]]	✗	6.177

7502	$y^{'} = x^{2} (1 + y^{2})$	[_separable]	✓	2.178

7503	$y^{'} = \frac{x^{2}}{1 - y^{2}}$	[_separable]	✓	1.115

7504	$y^{'} = \frac{3 x^{2} + 4 x + 2}{- 2 + 2 y}$ i.c.	[_separable]	✓	2.052

7505	$y^{'} x - 2 \sqrt{x y} = y$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	6.156

7506	$y^{'} = \frac{x + y - 1}{x - y + 3}$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.568

7507	$e^{x} + y + (x - 2 \sin (y)) y^{'} = 0$	[_exact]	✓	2.142

7508	$3 x + \frac{6}{y} + (\frac{x^{2}}{y} + \frac{3 y}{x}) y^{'} = 0$	[_rational]	✓	1.585

7509	$x^{2} y^{'} + y^{2} - x y = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	1.747

7510	$x + y - (x - y) y^{'} = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.506

7511	$y^{'} = \frac{y}{2 x} + \frac{x^{2}}{2 y}$	[[_homogeneous, ‘class G‘], _rational, _Bernoulli]	✓	2.020

7512	$y^{'} = - \frac{2}{t} + \frac{y}{t} + \frac{y^{2}}{t}$	[_separable]	✓	2.547

7513	$y^{'} = - \frac{y}{t} - 1 - y^{2}$	[_rational, _Riccati]	✓	1.437

7514	$y y^{'} + x = a {y^{'}}^{2}$	[_dAlembert]	✓	91.575

7515	${y^{'}}^{2} - a^{2} y^{2} = 0$	[_quadrature]	✓	0.605

7516	${y^{'}}^{2} = 4 x^{2}$	[_quadrature]	✓	0.376

7517	$y^{''} - 2 y^{'} - 3 y = 0$	[[_2nd_order, _missing_x]]	✓	0.340

7518	$s^{''} + 2 s^{'} + s = 0$ i.c.	[[_2nd_order, _missing_x]]	✓	0.660

7519	$y^{''} - 2 y^{'} + 5 y = 0$	[[_2nd_order, _missing_x]]	✓	0.504

7520	$y^{''} - 2 y^{'} - 3 y = 3 x + 1$	[[_2nd_order, _with_linear_symmetries]]	✓	0.439

7521	$y^{''} - 3 y^{'} + 2 y = x e^{2 x}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.464

7522	$y^{''} + y = 4 \sin (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.705

7523	$y^{''} + 2 x^{2} y^{'} + (x^{4} + 2 x - 1) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.667

7524	$p x^{2} u^{''} + q x u^{'} + r u = f (x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	3.707

7525	$\sin (x) u^{''} + 2 \cos (x) u^{'} + \sin (x) u = 0$	[_Lienard]	✓	0.908

7526	$3 {y^{''}}^{2} - y^{'''} y^{'} - y^{''} {y^{'}}^{2} = 0$	[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_exponential_symmetries], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]	✓	1.254

7527	$y^{''} - \frac{x y^{'}}{- x^{2} + 1} + \frac{y}{- x^{2} + 1} = 0$	[_Gegenbauer, [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	1.240

7528	$x^{2} y y^{''} = x^{2} {y^{'}}^{2} - y^{2}$	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]	✗	0.272

7529	$y^{'''} - 3 y^{''} + 3 y^{'} - y = 4 e^{t}$	[[_3rd_order, _with_linear_symmetries]]	✓	0.159

7530	$y^{''''} + 2 y^{''} + y = 3 \sin (t) - 5 \cos (t)$	[[_high_order, _linear, _nonhomogeneous]]	✓	1.246

7531	$y^{'''} - y^{''} - y^{'} + y = g (t)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	0.360

7532	$y^{(5)} - \frac{y^{''''}}{t} = 0$	[[_high_order, _missing_y]]	✓	0.883

7533	$x x^{''} - {x^{'}}^{2} = 0$	[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	0.509

7534	$y^{''''} + 4 y^{'''} + 3 y^{''} - 4 y^{'} - 4 y = f (x)$	[[_high_order, _linear, _nonhomogeneous]]	✓	0.439

7535	$u^{''} - (2 x + 1) u^{'} + (x^{2} + x - 1) u = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.569

7536	$y^{''} + 6 y^{'} + 9 y = 50 e^{2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.591

7537	$y^{''} - 4 y^{'} + 4 y = 50 e^{2 x}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.576

7538	$y^{''} + 3 y^{'} + 2 y = \cos (2 x)$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.573

7539	$y^{'''} + 6 y^{''} + 11 y^{'} + 6 y = 2 \sin (3 x)$	[[_3rd_order, _linear, _nonhomogeneous]]	✓	0.201

7540	$y^{''} + 4 y = x^{2}$	[[_2nd_order, _with_linear_symmetries]]	✓	0.607

7541	$y^{''} - 4 y^{'} + 3 y = x^{3}$	[[_2nd_order, _linear, _nonhomogeneous]]	✓	0.506

7542	$y^{''} + 2 y^{'} + (1 + \frac{2}{{(3 x + 1)}^{2}}) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.494

7543	$y + \sqrt{x^{2} + y^{2}} - y^{'} x = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	5.353

7544	${y^{'}}^{2} = a^{2} - y^{2}$	[_quadrature]	✓	1.026

7545	$x^{2} y^{''} - 2 y^{'} x + (x^{2} + 2) y = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.661

7546	$y^{''} + \frac{2 y^{'}}{x} - \frac{2 y}{{(x + 1)}^{2}} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.411

7547	$(1 + x^{2} y^{2}) y + (x^{2} y^{2} - 1) x y^{'} = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	2.169

7548	$2 x^{3} y^{2} - y + (2 x^{2} y^{3} - x) y^{'} = 0$	[_rational]	✓	1.666

7549	$\frac{1}{y} + \sec (\frac{y}{x}) - \frac{x y^{'}}{y^{2}} = 0$	[[_homogeneous, ‘class D‘]]	✓	16.407

7550	$ϕ^{'} - \frac{ϕ^{2}}{2} - ϕ \cot (θ) = 0$	[_Bernoulli]	✓	3.125

7551	$u^{''} - \cot (θ) u^{'} = 0$	[[_2nd_order, _missing_y]]	✓	0.533

7552	$(ϕ^{'} - \frac{ϕ^{2}}{2}) \sin {(θ)}^{2} - ϕ \sin (θ) \cos (θ) = \frac{\cos (2 θ)}{2} + 1$	[[_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], _Riccati]	✓	4.871

7553	$a y^{''} y^{'''} = \sqrt{1 + {y^{''}}^{2}}$	[[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]	✓	2.934

7554	$a^{2} y^{''''} = y^{''}$	[[_high_order, _missing_x]]	✓	0.082

7555	$y e^{x y} + x e^{x y} y^{'} = 0$	[_separable]	✓	1.620

7556	$x - 2 x y + e^{y} + (y - x^{2} + x e^{y}) y^{'} = 0$	[_exact]	✓	1.913

7557	$y^{''} - \frac{y^{'}}{\sqrt{x}} + \frac{(x + \sqrt{x} - 8) y}{4 x^{2}} = 0$	[[_2nd_order, _with_linear_symmetries]]	✓	0.519

7558	$(- x^{2} + 1) z^{''} + (1 - 3 x) z^{'} + k z = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.195

7559	$(- x^{2} + 1) η^{''} - (x + 1) η^{'} + (k + 1) η = 0$	[[_2nd_order, _with_linear_symmetries]]	✗	1.076

7560	$x^{2} + y^{2} - 2 x y^{'} y = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	7.243

7561	$x^{2} - y^{2} + 2 x y^{'} y = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	9.132

7562	$y^{'} x - y = x^{2} + y^{2}$	[[_homogeneous, ‘class D‘], _rational, _Riccati]	✓	1.610

7563	$y^{'} x - y = x \sqrt{x^{2} - y^{2}} y^{'}$	[‘y=_G(x,y’)‘]	✗	3.802

7564	$x + y y^{'} + y - y^{'} x = 0$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.501

7565	$y y^{''} - y^{2} y^{'} - {y^{'}}^{2} = 0$	[[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]	✓	0.701

7566	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - 18 x_{2} \\ x_{2}^{'} = 2 x_{1} - 9 x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.542

7567	$[\begin{matrix} x_{1}^{'} = x_{1} + 3 x_{2} \\ x_{2}^{'} = 5 x_{1} + 3 x_{2} \end{matrix}]$	system_of_ODEs	✓	0.404

7568	$[\begin{matrix} x_{1}^{'} = - x_{1} + 3 x_{2} \\ x_{2}^{'} = - 3 x_{1} + 5 x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.487

7569	$[\begin{matrix} x_{1}^{'} = 4 x_{1} - x_{2} \\ x_{2}^{'} = 5 x_{1} + 2 x_{2} \end{matrix}]$	system_of_ODEs	✓	0.553

7570	$[\begin{matrix} x_{1}^{'} = - 2 x_{1} + x_{2} \\ x_{2}^{'} = x_{1} - 2 x_{2} \end{matrix}]$	system_of_ODEs	✓	0.329

7571	$[\begin{matrix} x_{1}^{'} = - 2 x_{1} + x_{2} + 2 e^{- t} \\ x_{2}^{'} = x_{1} - 2 x_{2} + 3 t \end{matrix}]$	system_of_ODEs	✓	0.920

7572	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - x_{2} \\ x_{2}^{'} = 16 x_{1} - 5 x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.480

7573	$[\begin{matrix} x_{1}^{'} = x_{1} - 2 x_{2} \\ x_{2}^{'} = 3 x_{1} - 4 x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.503

7574	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - 18 x_{2} \\ x_{2}^{'} = 2 x_{1} - 9 x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.498

7575	$[\begin{matrix} x_{1}^{'} = - x_{1} + 3 x_{2} \\ x_{2}^{'} = - 3 x_{1} + 5 x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.469

7576	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - 18 x_{2} \\ x_{2}^{'} = 2 x_{1} - 9 x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.523

7577	$[\begin{matrix} x_{1}^{'} = 3 x_{1} - x_{2} \\ x_{2}^{'} = 4 x_{1} - 2 x_{2} \end{matrix}]$ i.c.	system_of_ODEs	✓	0.466

7578	$[\begin{matrix} x_{1}^{'} = x_{1} + x_{2} - 8 \\ x_{2}^{'} = x_{1} + x_{2} + 3 \end{matrix}]$	system_of_ODEs	✓	0.585

7579	$[\begin{matrix} x_{1}^{'} = x_{1} + x_{2} - 8 \\ x_{2}^{'} = x_{1} + x_{2} + 3 \end{matrix}]$ i.c.	system_of_ODEs	✓	0.653

7580	$y^{'} = e^{3 x} + \sin (x)$	[_quadrature]	✓	0.530

7581	$y^{''} = x + 2$	[[_2nd_order, _quadrature]]	✓	0.822

7582	$y^{'''} = x^{2}$	[[_3rd_order, _quadrature]]	✓	0.134

7583	$y^{'} + \cos (x) y = 0$	[_separable]	✓	1.466

7584	$y^{'} + \cos (x) y = \sin (x) \cos (x)$	[_linear]	✓	1.897

7585	$y^{''} - y = 0$	[[_2nd_order, _missing_x]]	✓	1.544

7586	$y^{''} + 4 y = 0$	[[_2nd_order, _missing_x]]	✓	1.457

7587	$y^{''} + k^{2} y = 0$	[[_2nd_order, _missing_x]]	✓	1.415

7588	$y^{'} + 5 y = 2$	[_quadrature]	✓	0.961

7589	$y^{''} = 3 x + 1$	[[_2nd_order, _quadrature]]	✓	0.867

7590	$y^{'} = k y$	[_quadrature]	✓	0.717

7591	$y^{'} - 2 y = 1$	[_quadrature]	✓	0.682

7592	$y^{'} + y = e^{x}$	[[_linear, ‘class A‘]]	✓	1.000

7593	$y^{'} - 2 y = x^{2} + x$	[[_linear, ‘class A‘]]	✓	1.131

7594	$y + 3 y^{'} = 2 e^{- x}$	[[_linear, ‘class A‘]]	✓	1.187

7595	$y^{'} + 3 y = e^{i x}$	[[_linear, ‘class A‘]]	✓	0.963

7596	$y^{'} + i y = x$	[[_linear, ‘class A‘]]	✓	1.046

7597	$L y^{'} + R y = E$	[_quadrature]	✓	0.762

7598	$L y^{'} + R y = E \sin (ω x)$ i.c.	[[_linear, ‘class A‘]]	✓	1.876

7599	$L y^{'} + R y = E e^{i ω x}$ i.c.	[[_linear, ‘class A‘]]	✓	1.586

7600	$y^{'} + a y = b (x)$	[[_linear, ‘class A‘]]	✓	1.101

2.2.76 Problems 7501 to 7600