Problems 4201 to 4300

Table 2.87: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	time (sec)

4201	$\cot (x) y^{'} + y = x$	[_linear]	✓	1.821

4202	$\cot (x) y^{'} + y = \tan (x)$	[_linear]	✓	2.262

4203	$y^{'} \tan (x) + y = \cot (x)$	[_linear]	✓	2.102

4204	$y^{'} \tan (x) = y - \cos (x)$	[_linear]	✓	2.821

4205	$y^{'} + \cos (x) y = \sin (2 x)$	[_linear]	✓	1.807

4206	$\cos (x) y^{'} + y = \sin (2 x)$	[_linear]	✓	2.848

4207	$y^{'} + y \sin (x) = \sin (2 x)$	[_linear]	✓	1.921

4208	$y^{'} \sin (x) + y = \sin (2 x)$	[_linear]	✓	2.895

4209	$\sqrt{x^{2} + 1} y^{'} + y = 2 x$	[_linear]	✓	1.744

4210	$\sqrt{x^{2} + 1} y^{'} - y = 2 \sqrt{x^{2} + 1}$	[_linear]	✓	1.895

4211	$\sqrt{(x + a) (x + b)} (2 y^{'} - 3) + y = 0$	[_linear]	✓	2.346

4212	$\sqrt{(x + a) (x + b)} y^{'} + y = \sqrt{x + a} - \sqrt{x + b}$	[_linear]	✓	3.324

4213	$3 y^{2} y^{'} = 2 x - 1$	[_separable]	✓	2.133

4214	$y^{'} = 6 x y^{2}$	[_separable]	✓	1.635

4215	$y^{'} = e^{y} \sin (x)$	[_separable]	✓	1.586

4216	$y^{'} = e^{x - y}$	[_separable]	✓	1.885

4217	$y^{'} = x \sec (y)$	[_separable]	✓	1.294

4218	$y^{'} = 3 \cos {(y)}^{2}$	[_quadrature]	✓	0.879

4219	$y^{'} x = y$	[_separable]	✓	1.426

4220	$(1 - x) y^{'} = y$	[_separable]	✓	1.406

4221	$y^{'} = \frac{4 x y}{x^{2} + 1}$	[_separable]	✓	1.210

4222	$y^{'} = \frac{2 y}{x^{2} - 1}$	[_separable]	✓	1.292

4223	$- y^{2} + x^{2} y^{'} = 0$ i.c.	[_separable]	✓	2.729

4224	$y^{'} + 2 x y = 0$ i.c.	[_separable]	✓	2.417

4225	$\cot (x) y^{'} = y$ i.c.	[_separable]	✓	1.989

4226	$y^{'} = x e^{- 2 y}$ i.c.	[_separable]	✓	1.716

4227	$y^{'} - 2 x y = 2 x$ i.c.	[_separable]	✓	1.428

4228	$y^{'} x = x y + y$ i.c.	[_separable]	✓	1.498

4229	$(x^{3} + 1) y^{'} = 3 x^{2} \tan (x)$ i.c.	[_quadrature]	✓	1.074

4230	$x \cos (y) y^{'} = 1 + \sin (y)$ i.c.	[_separable]	✓	2.999

4231	$y^{'} x = 2 y (- 1 + y)$ i.c.	[_separable]	✓	2.319

4232	$2 y^{'} x = 1 - y^{2}$ i.c.	[_separable]	✓	1.842

4233	$(1 - x) y^{'} = x y$	[_separable]	✓	1.336

4234	$(x^{2} - 1) y^{'} = (x^{2} + 1) y$	[_separable]	✓	1.653

4235	$y^{'} = e^{x} (1 + y^{2})$	[_separable]	✓	2.090

4236	$e^{y} y^{'} + 2 x = 2 x e^{y}$	[_separable]	✓	1.561

4237	$y e^{2 x} y^{'} + 2 x = 0$ i.c.	[_separable]	✓	2.144

4238	$x y^{'} y = \sqrt{y^{2} - 9}$ i.c.	[_separable]	✓	4.216

4239	$(x + y - 1) y^{'} = x - y + 1$	[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.871

4240	$x y^{'} y = 2 x^{2} - y^{2}$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	5.255

4241	$x^{2} - y^{2} + x y^{'} y = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	5.328

4242	$x^{2} y^{'} - 2 x y - 2 y^{2} = 0$	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	2.270

4243	$x^{2} y^{'} = 3 (x^{2} + y^{2}) \arctan (\frac{y}{x}) + x y$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	10.804

4244	$x \sin (\frac{y}{x}) y^{'} = y \sin (\frac{y}{x}) + x$	[[_homogeneous, ‘class A‘], _dAlembert]	✓	5.866

4245	$y^{'} x = y + 2 e^{- \frac{y}{x}}$	[[_homogeneous, ‘class D‘]]	✓	1.661

4246	$y^{'} = {(x + y)}^{2}$	[[_homogeneous, ‘class C‘], _Riccati]	✓	1.506

4247	$y^{'} = \sin {(x - y + 1)}^{2}$	[[_homogeneous, ‘class C‘], _dAlembert]	✓	6.173

4248	$y^{'} = \frac{x + y + 4}{x - y - 6}$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.564

4249	$y^{'} = \frac{x + y + 4}{x + y - 6}$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	1.962

4250	$(x + \frac{2}{y}) y^{'} + y = 0$	[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.483

4251	$\sin (x) \tan (y) + 1 + \cos (x) \sec {(y)}^{2} y^{'} = 0$	[‘y=_G(x,y’)‘]	✓	43.641

4252	$y - x^{3} + (y^{3} + x) y^{'} = 0$	[_exact, _rational]	✓	1.288

4253	$2 y^{2} - 4 x + 5 = (4 - 2 y + 4 x y) y^{'}$	[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✗	1.650

4254	$y + y \cos (x y) + (x + x \cos (x y)) y^{'} = 0$	[_separable]	✓	1.814

4255	$\cos (x) \cos {(y)}^{2} + 2 \sin (x) \sin (y) \cos (y) y^{'} = 0$	[_separable]	✓	2.938

4256	$(\sin (x) \sin (y) - x e^{y}) y^{'} = e^{y} + \cos (x) \cos (y)$	[_exact]	✓	31.104

4257	$- \frac{\sin (\frac{x}{y})}{y} + \frac{x \sin (\frac{x}{y}) y^{'}}{y^{2}} = 0$	[_separable]	✓	1.598

4258	$1 + y + (1 - x) y^{'} = 0$	[_separable]	✓	1.460

4259	$2 x y^{3} + \cos (x) y + (3 x^{2} y^{2} + \sin (x)) y^{'} = 0$	[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	43.661

4260	$1 = \frac{y}{1 - x^{2} y^{2}} + \frac{x y^{'}}{1 - x^{2} y^{2}}$	[_exact, _rational, _Riccati]	✓	1.580

4261	$(3 x^{2} - y^{2}) y^{'} - 2 x y = 0$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	3.497

4262	$x y - 1 + (x^{2} - x y) y^{'} = 0$	[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.315

4263	$(x + 3 y^{4} x^{3}) y^{'} + y = 0$	[[_homogeneous, ‘class G‘], _rational]	✓	12.710

4264	$(x - 1 - y^{2}) y^{'} - y = 0$	[[_1st_order, _with_linear_symmetries], _rational]	✓	1.555

4265	$y - (x + x y^{3}) y^{'} = 0$	[_separable]	✓	2.534

4266	$y^{'} x = x^{5} + x^{3} y^{2} + y$	[[_homogeneous, ‘class D‘], _rational, _Riccati]	✓	2.235

4267	$(x + y) y^{'} = y - x$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	3.489

4268	$y^{'} x = y + x^{2} + 9 y^{2}$	[[_homogeneous, ‘class D‘], _rational, _Riccati]	✓	1.375

4269	$y^{'} x - 3 y = x^{4}$	[_linear]	✓	1.514

4270	$y^{'} + y = \frac{1}{e^{2 x} + 1}$	[_linear]	✓	1.836

4271	$(x^{2} + 1) y^{'} + 2 x y = \cot (x)$	[_linear]	✓	1.717

4272	$y^{'} + y = 2 x e^{- x} + x^{2}$	[[_linear, ‘class A‘]]	✓	2.553

4273	$y^{'} + \cot (x) y = 2 x \csc (x)$	[_linear]	✓	1.784

4274	$2 y - x^{3} = y^{'} x$	[_linear]	✓	1.536

4275	$(1 - x y) y^{'} = y^{2}$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.514

4276	$2 x + 3 y + 1 + (2 y - 3 x + 5) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	2.495

4277	$y^{'} x = \sqrt{x^{2} + y^{2}}$	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	9.765

4278	$y^{2} = (x^{3} - x y) y^{'}$	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	1.954

4279	$x^{2} y^{3} + y = (x^{3} y^{2} - x) y^{'}$	[[_homogeneous, ‘class G‘], _rational]	✓	2.686

4280	$y^{'} x + y = x \cos (x)$	[_linear]	✓	1.323

4281	$(x y - x^{2}) y^{'} = y^{2}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	108.188

4282	$(e^{x} - 3 x^{2} y^{2}) y^{'} + y e^{x} = 2 x y^{3}$	[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]	✓	2.130

4283	$y + x^{2} = y^{'} x$	[_linear]	✓	1.192

4284	$y^{'} x + y = x^{2} \cos (x)$	[_linear]	✓	1.415

4285	$6 x + 4 y + 3 + (3 x + 2 y + 2) y^{'} = 0$	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	1.909

4286	$\cos (x + y) - x \sin (x + y) = x \sin (x + y) y^{'}$	[[_1st_order, _with_linear_symmetries], _exact]	✓	3.251

4287	$y^{2} e^{x y} + \cos (x) + (e^{x y} + x y e^{x y}) y^{'} = 0$	[_exact]	✓	36.130

4288	$y^{'} \ln (x - y) = 1 + \ln (x - y)$	[[_homogeneous, ‘class C‘], _exact, _dAlembert]	✓	1.803

4289	$y^{'} + 2 x y = e^{- x^{2}}$	[_linear]	✓	1.408

4290	$y^{2} - 3 x y - 2 x^{2} = (x^{2} - x y) y^{'}$	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	4.571

4291	$(x^{2} + 1) y^{'} + 2 x y = 4 x^{3}$	[_linear]	✓	1.600

4292	$e^{x} \sin (y) - y \sin (x y) + (e^{x} \cos (y) - x \sin (x y)) y^{'} = 0$	[_exact]	✓	41.731

4293	$(y - x^{2} + x e^{y}) y^{'} = 2 x y - e^{y} - x$	[_exact]	✓	1.901

4294	$e^{x} (x + 1) = (x e^{x} - e^{y} y) y^{'}$	[‘y=_G(x,y’)‘]	✓	2.006

4295	$x^{2} y^{'} + 2 x y = 0$	[_separable]	✓	1.590

4296	$x + y + (x - y) y^{'} = 0$ i.c.	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	1.348

4297	$\ln (x) y^{'} + \frac{x + y}{x} = 0$	[_linear]	✓	1.259

4298	$\cos (y) - x \sin (y) y^{'} = \sec {(x)}^{2}$ i.c.	[_exact]	✓	42.246

4299	$y \sin (\frac{x}{y}) + x \cos (\frac{x}{y}) - 1 + (x \sin (\frac{x}{y}) - \frac{x^{2} \cos (\frac{x}{y})}{y}) y^{'} = 0$	[_exact]	✓	40.116

4300	$\frac{x}{x^{2} + y^{2}} + \frac{y}{x^{2}} + (\frac{y}{x^{2} + y^{2}} - \frac{1}{x}) y^{'} = 0$ i.c.	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	221.105

2.2.43 Problems 4201 to 4300