2.2.43 Problems 4201 to 4300

Table 2.99: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

Maple

Mma

Sympy

time(sec)

4201

\begin{align*} \cot \left (x \right ) y^{\prime }+y&=\tan \left (x \right ) \\ \end{align*}

[_linear]

1.875

4202

\begin{align*} \tan \left (x \right ) y^{\prime }+y&=\cot \left (x \right ) \\ \end{align*}

[_linear]

1.863

4203

\begin{align*} \tan \left (x \right ) y^{\prime }&=y-\cos \left (x \right ) \\ \end{align*}

[_linear]

2.299

4204

\begin{align*} y^{\prime }+y \cos \left (x \right )&=\sin \left (2 x \right ) \\ \end{align*}

[_linear]

1.753

4205

\begin{align*} \cos \left (x \right ) y^{\prime }+y&=\sin \left (2 x \right ) \\ \end{align*}

[_linear]

2.625

4206

\begin{align*} y^{\prime }+y \sin \left (x \right )&=\sin \left (2 x \right ) \\ \end{align*}

[_linear]

1.783

4207

\begin{align*} \sin \left (x \right ) y^{\prime }+y&=\sin \left (2 x \right ) \\ \end{align*}

[_linear]

2.598

4208

\begin{align*} \sqrt {x^{2}+1}\, y^{\prime }+y&=2 x \\ \end{align*}

[_linear]

1.787

4209

\begin{align*} \sqrt {x^{2}+1}\, y^{\prime }-y&=2 \sqrt {x^{2}+1} \\ \end{align*}

[_linear]

2.086

4210

\begin{align*} \sqrt {\left (x +a \right ) \left (x +b \right )}\, \left (2 y^{\prime }-3\right )+y&=0 \\ \end{align*}

[_linear]

3.919

4211

\begin{align*} \sqrt {\left (x +a \right ) \left (x +b \right )}\, y^{\prime }+y&=\sqrt {x +a}-\sqrt {x +b} \\ \end{align*}

[_linear]

6.043

4212

\begin{align*} 3 y^{2} y^{\prime }&=2 x -1 \\ \end{align*}

[_separable]

1.815

4213

\begin{align*} y^{\prime }&=6 x y^{2} \\ \end{align*}

[_separable]

2.628

4214

\begin{align*} y^{\prime }&={\mathrm e}^{y} \sin \left (x \right ) \\ \end{align*}

[_separable]

1.473

4215

\begin{align*} y^{\prime }&={\mathrm e}^{x -y} \\ \end{align*}

[_separable]

1.221

4216

\begin{align*} y^{\prime }&=x \sec \left (y\right ) \\ \end{align*}

[_separable]

1.356

4217

\begin{align*} y^{\prime }&=3 \cos \left (y\right )^{2} \\ \end{align*}

[_quadrature]

0.336

4218

\begin{align*} y^{\prime } x&=y \\ \end{align*}

[_separable]

1.463

4219

\begin{align*} \left (1-x \right ) y^{\prime }&=y \\ \end{align*}

[_separable]

1.404

4220

\begin{align*} y^{\prime }&=\frac {4 x y}{x^{2}+1} \\ \end{align*}

[_separable]

1.599

4221

\begin{align*} y^{\prime }&=\frac {2 y}{x^{2}-1} \\ \end{align*}

[_separable]

1.603

4222

\begin{align*} x^{2} y^{\prime }-y^{2}&=0 \\ y \left (1\right ) &= -1 \\ \end{align*}

[_separable]

2.878

4223

\begin{align*} y^{\prime }+2 y x&=0 \\ y \left (0\right ) &= 5 \\ \end{align*}

[_separable]

1.683

4224

\begin{align*} \cot \left (x \right ) y^{\prime }&=y \\ y \left (0\right ) &= 2 \\ \end{align*}

[_separable]

2.993

4225

\begin{align*} y^{\prime }&=x \,{\mathrm e}^{-2 y} \\ y \left (0\right ) &= 0 \\ \end{align*}

[_separable]

3.191

4226

\begin{align*} y^{\prime }-2 y x&=2 x \\ y \left (0\right ) &= 1 \\ \end{align*}

[_separable]

2.641

4227

\begin{align*} y^{\prime } x&=y x +y \\ y \left (1\right ) &= 1 \\ \end{align*}

[_separable]

1.010

4228

\begin{align*} \left (x^{3}+1\right ) y^{\prime }&=3 x^{2} \tan \left (x \right ) \\ y \left (0\right ) &= \frac {\pi }{2} \\ \end{align*}

[_quadrature]

0.506

4229

\begin{align*} x \cos \left (y\right ) y^{\prime }&=1+\sin \left (y\right ) \\ y \left (1\right ) &= 0 \\ \end{align*}

[_separable]

4.845

4230

\begin{align*} y^{\prime } x&=2 y \left (y-1\right ) \\ y \left (\frac {1}{2}\right ) &= 2 \\ \end{align*}

[_separable]

3.565

4231

\begin{align*} 2 y^{\prime } x&=1-y^{2} \\ y \left (1\right ) &= 0 \\ \end{align*}

[_separable]

2.535

4232

\begin{align*} \left (1-x \right ) y^{\prime }&=y x \\ \end{align*}

[_separable]

1.829

4233

\begin{align*} \left (x^{2}-1\right ) y^{\prime }&=\left (x^{2}+1\right ) y \\ \end{align*}

[_separable]

2.049

4234

\begin{align*} y^{\prime }&={\mathrm e}^{x} \left (1+y^{2}\right ) \\ \end{align*}

[_separable]

2.424

4235

\begin{align*} {\mathrm e}^{y} y^{\prime }+2 x&=2 x \,{\mathrm e}^{y} \\ \end{align*}

[_separable]

1.769

4236

\begin{align*} y \,{\mathrm e}^{2 x} y^{\prime }+2 x&=0 \\ y \left (0\right ) &= 1 \\ \end{align*}

[_separable]

1.873

4237

\begin{align*} x y^{\prime } y&=\sqrt {y^{2}-9} \\ y \left ({\mathrm e}^{4}\right ) &= 5 \\ \end{align*}

[_separable]

8.641

4238

\begin{align*} \left (x +y-1\right ) y^{\prime }&=x -y+1 \\ \end{align*}

[[_homogeneous, ‘class C‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3.390

4239

\begin{align*} x y^{\prime } y&=2 x^{2}-y^{2} \\ \end{align*}

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

7.196

4240

\begin{align*} x^{2}-y^{2}+x y^{\prime } y&=0 \\ \end{align*}

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

4.287

4241

\begin{align*} x^{2} y^{\prime }-2 y x -2 y^{2}&=0 \\ \end{align*}

[[_homogeneous, ‘class A‘], _rational, _Bernoulli]

2.522

4242

\begin{align*} x^{2} y^{\prime }&=3 \left (y^{2}+x^{2}\right ) \arctan \left (\frac {y}{x}\right )+y x \\ \end{align*}

[[_homogeneous, ‘class A‘], _dAlembert]

9.674

4243

\begin{align*} x \sin \left (\frac {y}{x}\right ) y^{\prime }&=\sin \left (\frac {y}{x}\right ) y+x \\ \end{align*}

[[_homogeneous, ‘class A‘], _dAlembert]

3.874

4244

\begin{align*} y^{\prime } x&=y+2 \,{\mathrm e}^{-\frac {y}{x}} \\ \end{align*}

[[_homogeneous, ‘class D‘]]

2.564

4245

\begin{align*} y^{\prime }&=\left (x +y\right )^{2} \\ \end{align*}

[[_homogeneous, ‘class C‘], _Riccati]

1.133

4246

\begin{align*} y^{\prime }&=\sin \left (x -y+1\right )^{2} \\ \end{align*}

[[_homogeneous, ‘class C‘], _dAlembert]

2.764

4247

\begin{align*} y^{\prime }&=\frac {x +y+4}{x -y-6} \\ \end{align*}

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

15.247

4248

\begin{align*} y^{\prime }&=\frac {x +y+4}{x +y-6} \\ \end{align*}

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

4.944

4249

\begin{align*} \left (x +\frac {2}{y}\right ) y^{\prime }+y&=0 \\ \end{align*}

[[_homogeneous, ‘class G‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

4.442

4250

\begin{align*} \sin \left (x \right ) \tan \left (y\right )+1+\cos \left (x \right ) \sec \left (y\right )^{2} y^{\prime }&=0 \\ \end{align*}

[‘y=_G(x,y’)‘]

31.447

4251

\begin{align*} y-x^{3}+\left (y^{3}+x \right ) y^{\prime }&=0 \\ \end{align*}

[_exact, _rational]

1.480

4252

\begin{align*} 2 y^{2}-4 x +5&=\left (4-2 y+4 y x \right ) y^{\prime } \\ \end{align*}

[_rational, [_Abel, ‘2nd type‘, ‘class B‘]]

16.220

4253

\begin{align*} y+y \cos \left (y x \right )+\left (x +x \cos \left (y x \right )\right ) y^{\prime }&=0 \\ \end{align*}

[_separable]

0.089

4254

\begin{align*} \cos \left (x \right ) \cos \left (y\right )^{2}+2 \sin \left (x \right ) \sin \left (y\right ) \cos \left (y\right ) y^{\prime }&=0 \\ \end{align*}

[_separable]

0.277

4255

\begin{align*} \left (\sin \left (x \right ) \sin \left (y\right )-x \,{\mathrm e}^{y}\right ) y^{\prime }&={\mathrm e}^{y}+\cos \left (x \right ) \cos \left (y\right ) \\ \end{align*}

[_exact]

35.638

4256

\begin{align*} -\frac {\sin \left (\frac {x}{y}\right )}{y}+\frac {x \sin \left (\frac {x}{y}\right ) y^{\prime }}{y^{2}}&=0 \\ \end{align*}

[_separable]

2.067

4257

\begin{align*} 1+y+\left (1-x \right ) y^{\prime }&=0 \\ \end{align*}

[_separable]

1.710

4258

\begin{align*} 2 x y^{3}+y \cos \left (x \right )+\left (3 y^{2} x^{2}+\sin \left (x \right )\right ) y^{\prime }&=0 \\ \end{align*}

[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

36.487

4259

\begin{align*} 1&=\frac {y}{1-y^{2} x^{2}}+\frac {x y^{\prime }}{1-y^{2} x^{2}} \\ \end{align*}

[_exact, _rational, _Riccati]

4.260

4260

\begin{align*} \left (3 x^{2}-y^{2}\right ) y^{\prime }-2 y x&=0 \\ \end{align*}

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

5.173

4261

\begin{align*} y x -1+\left (x^{2}-y x \right ) y^{\prime }&=0 \\ \end{align*}

[_rational, [_1st_order, ‘_with_symmetry_[F(x),G(x)]‘], [_Abel, ‘2nd type‘, ‘class B‘]]

1.485

4262

\begin{align*} \left (x +3 y^{4} x^{3}\right ) y^{\prime }+y&=0 \\ \end{align*}

[[_homogeneous, ‘class G‘], _rational]

3.450

4263

\begin{align*} \left (x -1-y^{2}\right ) y^{\prime }-y&=0 \\ \end{align*}

[[_1st_order, _with_linear_symmetries], _rational]

1.639

4264

\begin{align*} y-\left (x +x y^{3}\right ) y^{\prime }&=0 \\ \end{align*}

[_separable]

2.652

4265

\begin{align*} y^{\prime } x&=x^{5}+x^{3} y^{2}+y \\ \end{align*}

[[_homogeneous, ‘class D‘], _rational, _Riccati]

1.936

4266

\begin{align*} \left (x +y\right ) y^{\prime }&=y-x \\ \end{align*}

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

3.899

4267

\begin{align*} y^{\prime } x&=y+x^{2}+9 y^{2} \\ \end{align*}

[[_homogeneous, ‘class D‘], _rational, _Riccati]

1.122

4268

\begin{align*} y^{\prime } x -3 y&=x^{4} \\ \end{align*}

[_linear]

1.003

4269

\begin{align*} y^{\prime }+y&=\frac {1}{{\mathrm e}^{2 x}+1} \\ \end{align*}

[_linear]

1.690

4270

\begin{align*} \left (x^{2}+1\right ) y^{\prime }+2 y x&=\cot \left (x \right ) \\ \end{align*}

[_linear]

1.703

4271

\begin{align*} y^{\prime }+y&=2 x \,{\mathrm e}^{-x}+x^{2} \\ \end{align*}

[[_linear, ‘class A‘]]

3.103

4272

\begin{align*} y^{\prime }+\cot \left (x \right ) y&=2 x \csc \left (x \right ) \\ \end{align*}

[_linear]

1.522

4273

\begin{align*} 2 y-x^{3}&=y^{\prime } x \\ \end{align*}

[_linear]

0.996

4274

\begin{align*} \left (-y x +1\right ) y^{\prime }&=y^{2} \\ \end{align*}

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3.915

4275

\begin{align*} 2 x +3 y+1+\left (2 y-3 x +5\right ) y^{\prime }&=0 \\ \end{align*}

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

15.125

4276

\begin{align*} y^{\prime } x&=\sqrt {y^{2}+x^{2}} \\ \end{align*}

[[_homogeneous, ‘class A‘], _rational, _dAlembert]

12.133

4277

\begin{align*} y^{2}&=\left (x^{3}-y x \right ) y^{\prime } \\ \end{align*}

[[_homogeneous, ‘class G‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

3.885

4278

\begin{align*} x^{2} y^{3}+y&=\left (x^{3} y^{2}-x \right ) y^{\prime } \\ \end{align*}

[[_homogeneous, ‘class G‘], _rational]

3.347

4279

\begin{align*} y^{\prime } x +y&=\cos \left (x \right ) x \\ \end{align*}

[_linear]

1.425

4280

\begin{align*} \left (y x -x^{2}\right ) y^{\prime }&=y^{2} \\ \end{align*}

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

9.531

4281

\begin{align*} \left ({\mathrm e}^{x}-3 y^{2} x^{2}\right ) y^{\prime }+{\mathrm e}^{x} y&=2 x y^{3} \\ \end{align*}

[_exact, [_1st_order, ‘_with_symmetry_[F(x),G(x)*y+H(x)]‘]]

2.530

4282

\begin{align*} y+x^{2}&=y^{\prime } x \\ \end{align*}

[_linear]

1.066

4283

\begin{align*} y^{\prime } x +y&=x^{2} \cos \left (x \right ) \\ \end{align*}

[_linear]

1.648

4284

\begin{align*} 6 x +4 y+3+\left (3 x +2 y+2\right ) y^{\prime }&=0 \\ \end{align*}

[[_homogeneous, ‘class C‘], _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

5.066

4285

\begin{align*} \cos \left (x +y\right )-x \sin \left (x +y\right )&=x \sin \left (x +y\right ) y^{\prime } \\ \end{align*}

[[_1st_order, _with_linear_symmetries], _exact]

2.945

4286

\begin{align*} y^{2} {\mathrm e}^{y x}+\cos \left (x \right )+\left ({\mathrm e}^{y x}+x y \,{\mathrm e}^{y x}\right ) y^{\prime }&=0 \\ \end{align*}

[_exact]

28.667

4287

\begin{align*} y^{\prime } \ln \left (x -y\right )&=1+\ln \left (x -y\right ) \\ \end{align*}

[[_homogeneous, ‘class C‘], _exact, _dAlembert]

1.513

4288

\begin{align*} y^{\prime }+2 y x&={\mathrm e}^{-x^{2}} \\ \end{align*}

[_linear]

1.875

4289

\begin{align*} y^{2}-3 y x -2 x^{2}&=\left (x^{2}-y x \right ) y^{\prime } \\ \end{align*}

[[_homogeneous, ‘class A‘], _rational, [_Abel, ‘2nd type‘, ‘class B‘]]

5.410

4290

\begin{align*} \left (x^{2}+1\right ) y^{\prime }+2 y x&=4 x^{3} \\ \end{align*}

[_linear]

1.961

4291

\begin{align*} {\mathrm e}^{x} \sin \left (y\right )-y \sin \left (y x \right )+\left ({\mathrm e}^{x} \cos \left (y\right )-x \sin \left (y x \right )\right ) y^{\prime }&=0 \\ \end{align*}

[_exact]

4.616

4292

\begin{align*} \left (x \,{\mathrm e}^{y}+y-x^{2}\right ) y^{\prime }&=2 y x -{\mathrm e}^{y}-x \\ \end{align*}

[_exact]

2.197

4293

\begin{align*} {\mathrm e}^{x} \left (x +1\right )&=\left (x \,{\mathrm e}^{x}-{\mathrm e}^{y} y\right ) y^{\prime } \\ \end{align*}

[‘y=_G(x,y’)‘]

2.062

4294

\begin{align*} x^{2} y^{\prime }+2 y x&=0 \\ \end{align*}

[_separable]

2.116

4295

\begin{align*} x +y+\left (x -y\right ) y^{\prime }&=0 \\ y \left (0\right ) &= 0 \\ \end{align*}

[[_homogeneous, ‘class A‘], _exact, _rational, [_Abel, ‘2nd type‘, ‘class A‘]]

0.638

4296

\begin{align*} \ln \left (x \right ) y^{\prime }+\frac {x +y}{x}&=0 \\ \end{align*}

[_linear]

1.487

4297

\begin{align*} \cos \left (y\right )-x \sin \left (y\right ) y^{\prime }&=\sec \left (x \right )^{2} \\ y \left (0\right ) &= 0 \\ \end{align*}

[_exact]

33.135

4298

\begin{align*} y \sin \left (\frac {x}{y}\right )+x \cos \left (\frac {x}{y}\right )-1+\left (x \sin \left (\frac {x}{y}\right )-\frac {x^{2} \cos \left (\frac {x}{y}\right )}{y}\right ) y^{\prime }&=0 \\ \end{align*}

[_exact]

3.302

4299

\begin{align*} \frac {x}{y^{2}+x^{2}}+\frac {y}{x^{2}}+\left (\frac {y}{y^{2}+x^{2}}-\frac {1}{x}\right ) y^{\prime }&=0 \\ y \left (1\right ) &= 0 \\ \end{align*}

[[_homogeneous, ‘class A‘], _exact, _rational, _dAlembert]

0.299

4300

\begin{align*} x^{2} \left (1+y^{2}\right ) y^{\prime }+y^{2} \left (x^{2}+1\right )&=0 \\ \end{align*}

[_separable]

2.068