Problems 20201 to 20300

2.2.203 Problems 20201 to 20300

Table 2.419: Main lookup table. Sorted sequentially by problem number.


#	ODE	CAS classiﬁcation	Solved?	Maple	Mma	Sympy	time(sec)

20201	\begin{align} \left (x^{3}-x \right ) y^{\prime \prime }+y^{\prime }+n^{2} x^{3} y&=0 \\ \end{align}	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✗	86.039

20202	\begin{align} \left (-y+y^{\prime } x \right )^{2}+x^{2} y y^{\prime \prime }&=0 \\ \end{align}	[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]	✗	✓	✓	✗	0.539

20203	\begin{align} 1+{y^{\prime }}^{2}+y y^{\prime \prime }&=0 \\ \end{align}	[[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]	✓	✓	✓	✗	1.818

20204	\begin{align} x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }+n^{2} y&=0 \\ \end{align}	[[_Emden, _Fowler], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]	✓	✓	✓	✓	4.180

20205	\begin{align} x^{\prime }+2 x+y^{\prime }+y&=0 \\ 5 x+y^{\prime }+3 y&=0 \\ \end{align}	system_of_ODEs	✓	✓	✓	✓	0.427

20206	\begin{align} x^{\prime }-7 x+y&=0 \\ y^{\prime }-2 x-5 y&=0 \\ \end{align}	system_of_ODEs	✓	✓	✓	✓	0.553

20207	\begin{align} x^{\prime }+2 x-3 y&=t \\ y^{\prime }-3 x+2 y&={\mathrm e}^{2 t} \\ \end{align}	system_of_ODEs	✓	✓	✓	✓	0.804

20208	\begin{align} 4 x^{\prime }+9 y^{\prime }+44 x+49 y&=t \\ 3 x^{\prime }+7 y^{\prime }+34 x+38 y&={\mathrm e}^{t} \\ \end{align}	system_of_ODEs	✓	✓	✓	✓	0.755

20209	\begin{align} x^{\prime \prime }-3 x-4 y&=0 \\ x+y^{\prime \prime }+y&=0 \\ \end{align}	system_of_ODEs	✗	✓	✓	✓	0.029

20210	\begin{align} x^{\prime }+2 y^{\prime }-2 x+2 y&=3 \,{\mathrm e}^{t} \\ 3 x^{\prime }+y^{\prime }+2 x+y&=4 \,{\mathrm e}^{2 t} \\ \end{align}	system_of_ODEs	✓	✓	✓	✓	0.684

20211	\begin{align} 4 x^{\prime }+9 y^{\prime }+2 x+31 y&={\mathrm e}^{t} \\ 3 x^{\prime }+7 y^{\prime }+x+24 y&=3 \\ \end{align}	system_of_ODEs	✓	✓	✓	✓	1.027

20212	\begin{align} x^{\prime }+4 x+3 y&=t \\ y^{\prime }+2 x+5 y&={\mathrm e}^{t} \\ \end{align}	system_of_ODEs	✓	✓	✓	✓	0.645

20213	\begin{align} x^{\prime }&=n y-m z \\ y^{\prime }&=L z-m x \\ z^{\prime }&=m x-L y \\ \end{align}	system_of_ODEs	✓	✓	✓	✗	41.951

20214	\begin{align} x^{2} y^{\prime \prime }-5 y^{\prime } x +5 y&=\frac {1}{x} \\ \end{align}	[[_2nd_order, _with_linear_symmetries]]	✓	✓	✓	✓	4.982

20215	\begin{align} y^{\prime \prime }+\frac {2 y^{\prime }}{r}&=0 \\ \end{align}	[[_2nd_order, _missing_y]]	✓	✓	✓	✓	8.621

20216	\begin{align} y^{\prime } x +x +y&=0 \\ \end{align}	[_linear]	✓	✓	✓	✓	2.839

20217	\begin{align} \left (y x +1\right ) y-y^{\prime } x&=0 \\ \end{align}	[[_homogeneous, ‘class D‘], _rational, _Bernoulli]	✓	✓	✓	✓	3.641

20218	\begin{align} \sin \left (x \right ) y^{\prime }-\cos \left (x \right ) y+y^{2}&=0 \\ \end{align}	[_Bernoulli]	✓	✓	✓	✓	2.760

20219	\begin{align} y-x +\left (x +y\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	4.732

20220	\begin{align} x +y y^{\prime }+\frac {-y+y^{\prime } x}{x^{2}+y^{2}}&=0 \\ \end{align}	[[_1st_order, _with_linear_symmetries], _exact, _rational]	✓	✓	✓	✗	2.438

20221	\begin{align} x^{3}+3 x y^{2}+\left (y^{3}+3 x^{2} y\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	✓	✓	✓	10.734

20222	\begin{align} y^{\prime } x +y&=y^{2} \ln \left (x \right ) \\ \end{align}	[_Bernoulli]	✓	✓	✓	✓	4.289

20223	\begin{align} \left (-x^{2}+1\right ) y^{\prime }-2 y x&=-x^{3}+x \\ \end{align}	[_linear]	✓	✓	✓	✓	2.325

20224	\begin{align} y^{\prime } x -y-\cos \left (\frac {1}{x}\right )&=0 \\ \end{align}	[_linear]	✓	✓	✓	✓	1.953

20225	\begin{align} y y^{\prime }+x&=m \left (-y+y^{\prime } x \right ) \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✗	11.531

20226	\begin{align} x \cos \left (y\right )^{2}&=y \cos \left (x \right )^{2} y^{\prime } \\ \end{align}	[_separable]	✓	✓	✓	✓	4.566

20227	\begin{align} y^{\prime }&={\mathrm e}^{x -y}+x^{2} {\mathrm e}^{-y} \\ \end{align}	[_separable]	✓	✓	✓	✓	1.945

20228	\begin{align} x^{2} y^{\prime }+y&=1 \\ \end{align}	[_separable]	✓	✓	✓	✓	1.859

20229	\begin{align} 2 y+\left (x^{2}+1\right ) \arctan \left (x \right ) y^{\prime }&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	2.295

20230	\begin{align} x y^{2}+x +\left (y+x^{2} y\right ) y^{\prime }&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	2.266

20231	\begin{align} y^{\prime }&={\mathrm e}^{x +y}+x^{2} {\mathrm e}^{y} \\ \end{align}	[_separable]	✓	✓	✓	✓	1.969

20232	\begin{align} \left (3+2 \sin \left (x \right )+\cos \left (x \right )\right ) y^{\prime }&=1+2 \sin \left (y\right )+\cos \left (y\right ) \\ \end{align}	[_separable]	✓	✓	✓	✓	8.070

20233	\begin{align} \frac {\cos \left (y\right )^{2} y^{\prime }}{x}+\frac {\cos \left (x \right )^{2}}{y}&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	4.431

20234	\begin{align} \left ({\mathrm e}^{x}+1\right ) y y^{\prime }&=\left (1+y\right ) {\mathrm e}^{x} \\ \end{align}	[_separable]	✓	✓	✓	✓	3.187

20235	\begin{align} \csc \left (x \right ) \ln \left (y\right ) y^{\prime }+y^{2} x^{2}&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	5.171

20236	\begin{align} y^{\prime }&=\frac {\sin \left (x \right )+\cos \left (x \right ) x}{y \left (2 \ln \left (y\right )+1\right )} \\ \end{align}	[_separable]	✓	✓	✓	✓	0.815

20237	\begin{align} \cos \left (y\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )&=\cos \left (x \right ) \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) y^{\prime } \\ \end{align}	[_separable]	✓	✓	✓	✓	37.778

20238	\begin{align} y^{2}+x y^{2}+\left (x^{2}-x^{2} y\right ) y^{\prime }&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	3.708

20239	\begin{align} \left (\sin \left (y\right )+y \cos \left (y\right )\right ) y^{\prime }-\left (2 \ln \left (x \right )+1\right ) x&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	30.597

20240	\begin{align} 3 \,{\mathrm e}^{x} \tan \left (y\right )+\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime }&=0 \\ \end{align}	[_separable]	✓	✓	✓	✓	3.459

20241	\begin{align} -y^{\prime } x +y&=a \left (y^{\prime }+y^{2}\right ) \\ \end{align}	[_separable]	✓	✓	✓	✓	3.670

20242	\begin{align} \left (x +y-1\right ) y^{\prime }&=x +y+1 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	5.025

20243	\begin{align} \left (2 x +2 y+1\right ) y^{\prime }&=x +y+1 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	5.309

20244	\begin{align} 2 x +3 y-1+\left (2 x +3 y-5\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	5.687

20245	\begin{align} \left (x^{2}+y^{2}\right ) y^{\prime }&=y x +x^{2} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓	8.403

20246	\begin{align} \left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y-\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✗	7.709

20247	\begin{align} x^{2}-y^{2}+2 y y^{\prime } x&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	6.433

20248	\begin{align} y^{\prime }&=\frac {y}{x}+\tan \left (\frac {y}{x}\right ) \\ \end{align}	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✓	4.846

20249	\begin{align} \left (2 x -2 y+5\right ) y^{\prime }-x +y-3&=0 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	6.072

20250	\begin{align} x +y+1-\left (2 x +2 y+1\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	5.290

20251	\begin{align} y^{2}&=\left (y x -x^{2}\right ) y^{\prime } \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✓	13.214

20252	\begin{align} x \sin \left (\frac {y}{x}\right ) y^{\prime }&=y \sin \left (\frac {y}{x}\right )-x \\ \end{align}	[[_homogeneous, ‘class A‘], _dAlembert]	✓	✓	✓	✓	4.771

20253	\begin{align} \left (x^{2}+y^{2}\right ) y^{\prime }&=y x \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓	4.769

20254	\begin{align} x^{2} y^{\prime }+y \left (x +y\right )&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	5.726

20255	\begin{align} 2 y^{\prime }&=\frac {y}{x}+\frac {y^{2}}{x^{2}} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	12.434

20256	\begin{align} \left (6 x -5 y+4\right ) y^{\prime }+y-2 x -1&=0 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✗	39.018

20257	\begin{align} \left (x -3 y+4\right ) y^{\prime }+7 y-5 x&=0 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	38.447

20258	\begin{align} \left (3+2 x +4 y\right ) y^{\prime }&=x +2 y+1 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	6.112

20259	\begin{align} -y+y^{\prime } x&=\sqrt {x^{2}+y^{2}} \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _dAlembert]	✓	✓	✓	✓	8.573

20260	\begin{align} \left (3 x^{2}+y^{2}\right ) y y^{\prime }+x \left (x^{2}+3 y^{2}\right )&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]	✓	✓	✓	✓	8.413

20261	\begin{align} x^{2}+3 y^{2}-2 y y^{\prime } x&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	6.169

20262	\begin{align} y^{\prime }&=\frac {1+2 x -y}{x +2 y-3} \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	4.950

20263	\begin{align} \left (x -y\right ) y^{\prime }&=x +y+1 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	6.989

20264	\begin{align} x -y-2-\left (2 x -2 y-3\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]	✓	✓	✓	✓	6.323

20265	\begin{align} y^{\prime }+\cot \left (x \right ) y&=2 \cos \left (x \right ) \\ \end{align}	[_linear]	✓	✓	✓	✓	2.075

20266	\begin{align} \cos \left (x \right )^{2} y^{\prime }+y&=\tan \left (x \right ) \\ \end{align}	[_linear]	✓	✓	✓	✓	4.114

20267	\begin{align} x \cos \left (x \right ) y^{\prime }+y \left (x \sin \left (x \right )+\cos \left (x \right )\right )&=1 \\ \end{align}	[_linear]	✓	✓	✓	✓	6.469

20268	\begin{align} y-x \sin \left (x^{2}\right )+y^{\prime } x&=0 \\ \end{align}	[_linear]	✓	✓	✓	✓	1.851

20269	\begin{align} x \ln \left (x \right ) y^{\prime }+y&=2 \ln \left (x \right ) \\ \end{align}	[_linear]	✓	✓	✓	✓	2.136

20270	\begin{align} \cos \left (x \right ) \sin \left (x \right ) y^{\prime }&=y+\sin \left (x \right ) \\ \end{align}	[_linear]	✓	✓	✓	✗	3.565

20271	\begin{align} \left (x y^{2}+1+x \right ) y^{\prime }+y+y^{3}&=0 \\ \end{align}	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	✓	✓	✓	1.971

20272	\begin{align} y^{2}+\left (x -\frac {1}{y}\right ) y^{\prime }&=0 \\ \end{align}	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class C‘]]	✓	✓	✓	✓	5.968

20273	\begin{align} y^{\prime }+3 x^{2} y&=x^{5} {\mathrm e}^{x^{3}} \\ \end{align}	[_linear]	✓	✓	✓	✓	1.838

20274	\begin{align} y^{\prime }-\frac {\tan \left (y\right )}{x +1}&=\left (x +1\right ) {\mathrm e}^{x} \sec \left (y\right ) \\ \end{align}	[‘y=_G(x,y’)‘]	✗	✓	✓	✓	10.594

20275	\begin{align} y^{\prime }+\frac {\left (1-2 x \right ) y}{x^{2}}&=1 \\ \end{align}	[_linear]	✓	✓	✓	✓	2.484

20276	\begin{align} y^{\prime }+\frac {2 y}{x}&=\sin \left (x \right ) \\ \end{align}	[_linear]	✓	✓	✓	✓	1.867

20277	\begin{align} 1+y^{2}&=\left (\arctan \left (y\right )-x \right ) y^{\prime } \\ \end{align}	[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	✓	✓	✓	6.192

20278	\begin{align} 1+y+x^{2} y+\left (x^{3}+x \right ) y^{\prime }&=0 \\ \end{align}	[_linear]	✓	✓	✓	✓	1.641

20279	\begin{align} y^{\prime }+\frac {x y}{x^{2}+1}&=\frac {1}{2 x \left (x^{2}+1\right )} \\ \end{align}	[_linear]	✓	✓	✓	✓	1.648

20280	\begin{align} y^{\prime }+\frac {\tan \left (y\right )}{x}&=\frac {\tan \left (y\right ) \sin \left (y\right )}{x^{2}} \\ \end{align}	[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]	✓	✓	✓	✓	38.021

20281	\begin{align} y^{\prime }+\frac {y \ln \left (y\right )}{x}&=\frac {y}{x^{2}}-\ln \left (y\right )^{2} \\ \end{align}	[‘x=_G(y,y’)‘]	✗	✗	✗	✗	2.620

20282	\begin{align} x +y^{\prime }&=x \,{\mathrm e}^{\left (n -1\right ) y} \\ \end{align}	[_separable]	✓	✓	✓	✗	3.374

20283	\begin{align} y \left ({\mathrm e}^{x}+2 y x \right )-{\mathrm e}^{x} y^{\prime }&=0 \\ \end{align}	[_Bernoulli]	✓	✓	✓	✓	4.308

20284	\begin{align} 2 y^{\prime }-y \sec \left (x \right )&=y^{3} \tan \left (x \right ) \\ \end{align}	[_Bernoulli]	✓	✓	✓	✓	10.404

20285	\begin{align} y^{\prime }+\cos \left (x \right ) y&=y^{n} \sin \left (2 x \right ) \\ \end{align}	[_Bernoulli]	✓	✓	✓	✓	5.944

20286	\begin{align} y y^{\prime }+x&=\frac {a^{2} \left (-y+y^{\prime } x \right )}{x^{2}+y^{2}} \\ \end{align}	[[_1st_order, _with_linear_symmetries], _exact, _rational]	✓	✓	✓	✗	3.122

20287	\begin{align} 1+4 y x +2 y^{2}+\left (1+4 y x +2 x^{2}\right ) y^{\prime }&=0 \\ \end{align}	[_exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✗	2.658

20288	\begin{align} x^{2} y-2 x y^{2}-\left (x^{3}-3 x^{2} y\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✓	7.286

20289	\begin{align} \left (y^{4} x^{4}+y^{2} x^{2}+y x \right ) y+\left (y^{4} x^{4}-y^{2} x^{2}+y x \right ) x y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class G‘], _rational]	✓	✓	✓	✓	0.537

20290	\begin{align} y \left (y x +2 y^{2} x^{2}\right )+x \left (y x -y^{2} x^{2}\right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✓	0.485

20291	\begin{align} y^{4}+2 y+\left (x y^{3}+2 y^{4}-4 x \right ) y^{\prime }&=0 \\ \end{align}	[_rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]	✓	✓	✓	✓	3.090

20292	\begin{align} x^{2}+y^{2}-2 y y^{\prime } x&=0 \\ \end{align}	[[_homogeneous, ‘class A‘], _rational, _Bernoulli]	✓	✓	✓	✓	7.189

20293	\begin{align} \left (20 x^{2}+8 y x +4 y^{2}+3 y^{3}\right ) y+4 \left (x^{2}+y x +y^{2}+y^{3}\right ) x y^{\prime }&=0 \\ \end{align}	[_rational]	✓	✓	✓	✗	2.596

20294	\begin{align} y^{2}+2 x^{2} y+\left (2 x^{3}-y x \right ) y^{\prime }&=0 \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✗	5.636

20295	\begin{align} 2 y+3 y^{\prime } x +2 x y \left (3 y+4 y^{\prime } x \right )&=0 \\ \end{align}	[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]	✓	✓	✓	✗	16.344

20296	\begin{align} \frac {y y^{\prime }+x}{-y+y^{\prime } x}&=\sqrt {\frac {a^{2}-x^{2}-y^{2}}{x^{2}+y^{2}}} \\ \end{align}	[[_1st_order, _with_linear_symmetries]]	✗	✓	✓	✗	78.743

20297	\begin{align} \frac {\left (x +y-a \right ) y^{\prime }}{x +y-b}&=\frac {x +y+a}{x +y+b} \\ \end{align}	[[_homogeneous, ‘class C‘], _rational, _dAlembert]	✓	✓	✓	✓	2.711

20298	\begin{align} \left (x -y\right )^{2} y^{\prime }&=a^{2} \\ \end{align}	[[_homogeneous, ‘class C‘], _dAlembert]	✓	✓	✓	✓	3.349

20299	\begin{align} \left (x +y\right )^{2} y^{\prime }&=a^{2} \\ \end{align}	[[_homogeneous, ‘class C‘], _dAlembert]	✓	✓	✓	✓	6.819

20300	\begin{align} y^{\prime }&=\left (4 x +y+1\right )^{2} \\ \end{align}	[[_homogeneous, ‘class C‘], _Riccati]	✓	✓	✓	✓	6.683

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