2.2.43 Problems 4201 to 4300

Table 2.103: 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]

4.207

4202

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

[_linear]

4.128

4203

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

[_linear]

4.441

4204

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

[_linear]

3.174

4205

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

[_linear]

4.719

4206

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

[_linear]

3.233

4207

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

[_linear]

4.615

4208

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

[_linear]

4.701

4209

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

[_linear]

4.480

4210

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

[_linear]

6.672

4211

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

[_linear]

7.401

4212

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

[_separable]

4.202

4213

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

[_separable]

8.284

4214

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

[_separable]

2.965

4215

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

[_separable]

3.295

4216

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

[_separable]

2.704

4217

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

[_quadrature]

0.863

4218

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

[_separable]

3.412

4219

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

[_separable]

3.702

4220

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

[_separable]

3.892

4221

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

[_separable]

3.978

4222

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

[_separable]

8.166

4223

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

[_separable]

4.372

4224

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

[_separable]

6.964

4225

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

[_separable]

5.264

4226

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

[_separable]

3.963

4227

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

[_separable]

3.204

4228

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

[_quadrature]

1.339

4229

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

[_separable]

8.933

4230

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

[_separable]

12.881

4231

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

[_separable]

6.644

4232

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

[_separable]

4.869

4233

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

[_separable]

5.344

4234

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

[_separable]

5.006

4235

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

[_separable]

3.694

4236

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

[_separable]

3.954

4237

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

[_separable]

14.737

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‘]]

19.694

4239

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

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

18.926

4240

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

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

14.314

4241

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

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

7.348

4242

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

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

14.914

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]

33.927

4244

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

[[_homogeneous, ‘class D‘]]

5.405

4245

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

[[_homogeneous, ‘class C‘], _Riccati]

2.839

4246

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

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

6.147

4247

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

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

23.691

4248

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

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

10.762

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‘]]

57.701

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’)‘]

31.826

4251

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

[_exact, _rational]

3.210

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‘]]

23.849

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]

0.327

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]

0.988

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]

29.570

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]

4.712

4257

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

[_separable]

4.270

4258

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

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

30.529

4259

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

[_exact, _rational, _Riccati]

6.575

4260

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

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

15.815

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‘]]

5.572

4262

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

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

7.816

4263

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

[[_1st_order, _with_linear_symmetries], _rational]

3.826

4264

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

[_separable]

8.488

4265

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

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

4.530

4266

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

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

17.873

4267

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

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

3.055

4268

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

[_linear]

2.585

4269

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

[_linear]

2.964

4270

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

[_linear]

3.368

4271

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

[[_linear, ‘class A‘]]

5.499

4272

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

[_linear]

3.274

4273

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

[_linear]

2.947

4274

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

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

49.985

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‘]]

25.625

4276

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

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

29.806

4277

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

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

52.680

4278

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

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

11.391

4279

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

[_linear]

2.610

4280

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

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

28.250

4281

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

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

4.748

4282

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

[_linear]

3.009

4283

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

[_linear]

3.069

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‘]]

14.415

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]

6.687

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]

26.907

4287

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

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

4.550

4288

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

[_linear]

3.520

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‘]]

28.266

4290

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

[_linear]

3.546

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]

7.562

4292

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

[_exact]

3.552

4293

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

[‘y=_G(x,y’)‘]

3.799

4294

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

[_separable]

5.612

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‘]]

3.306

4296

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

[_linear]

2.810

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]

31.059

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]

10.980

4299

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

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

1.269

4300

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

[_separable]

4.175