2.2.35 Problems 3401 to 3500

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

#

ODE

CAS classification

Solved?

Maple

Mma

Sympy

time(sec)

3401

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

Series expansion around \(x=0\).

[[_2nd_order, _linear, _nonhomogeneous]]

1.243

3402

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

[_quadrature]

0.845

3403

\begin{align*} y^{\prime }&=2 \,{\mathrm e}^{3 x} \\ \end{align*}

[_quadrature]

0.399

3404

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

[_quadrature]

0.462

3405

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

[_quadrature]

0.490

3406

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

[_quadrature]

0.453

3407

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

[_quadrature]

0.428

3408

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

[_separable]

2.871

3409

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

[_separable]

7.214

3410

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

[_separable]

2.993

3411

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

[_separable]

2.542

3412

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

[_separable]

6.028

3413

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

[_quadrature]

0.572

3414

\begin{align*} {y^{\prime }}^{2}-3 y^{\prime }+2&=0 \\ \end{align*}

[_quadrature]

0.263

3415

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

[_quadrature]

0.447

3416

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

[_quadrature]

0.843

3417

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

[_quadrature]

0.378

3418

\begin{align*} y^{\prime }&={\mathrm e}^{2 t} t \\ \end{align*}

[_quadrature]

0.398

3419

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

[_quadrature]

0.448

3420

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

[_quadrature]

0.461

3421

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

[_quadrature]

0.477

3422

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

[_quadrature]

0.448

3423

\begin{align*} y^{\prime }&=\frac {t}{\sqrt {t}+1} \\ \end{align*}

[_quadrature]

0.491

3424

\begin{align*} y^{\prime }&=2 y-4 \\ y \left (0\right ) &= 5 \\ \end{align*}

[_quadrature]

1.474

3425

\begin{align*} y^{\prime }&=-y^{3} \\ y \left (1\right ) &= 3 \\ \end{align*}

[_quadrature]

8.155

3426

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{t}}{y} \\ y \left (\ln \left (2\right )\right ) &= -8 \\ \end{align*}

[_separable]

2.726

3427

\begin{align*} y^{\prime }&={\mathrm e}^{2 t} t \\ y \left (1\right ) &= 5 \\ \end{align*}

[_quadrature]

0.512

3428

\begin{align*} y^{\prime }&=\sin \left (t \right )^{2} \\ y \left (\frac {\pi }{6}\right ) &= 3 \\ \end{align*}

[_quadrature]

0.591

3429

\begin{align*} y^{\prime }&=8 \,{\mathrm e}^{4 t}+t \\ y \left (0\right ) &= 12 \\ \end{align*}

[_quadrature]

0.605

3430

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

[_separable]

2.665

3431

\begin{align*} y^{\prime }&=-\frac {t}{y} \\ \end{align*}

[_separable]

6.750

3432

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

[_quadrature]

1.369

3433

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

[_quadrature]

0.706

3434

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

[_quadrature]

0.802

3435

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

[_quadrature]

9.735

3436

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

[_quadrature]

2.560

3437

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

[_separable]

3.330

3438

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

[_quadrature]

1.112

3439

\begin{align*} y^{\prime }&=2 y+{\mathrm e}^{-3 t} \\ \end{align*}

[[_linear, ‘class A‘]]

1.724

3440

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

[[_linear, ‘class A‘]]

1.492

3441

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

[[_linear, ‘class A‘]]

1.408

3442

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

[_linear]

2.351

3443

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

[_linear]

2.221

3444

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

[_linear]

3.053

3445

\begin{align*} y^{\prime }&=\tan \left (t \right ) y+\sec \left (t \right )^{3} \\ \end{align*}

[_linear]

2.350

3446

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

[_quadrature]

1.222

3447

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

[_quadrature]

1.678

3448

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

[_linear]

3.604

3449

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

[_linear]

2.664

3450

\begin{align*} y^{\prime }&=\frac {2 y}{t +1} \\ y \left (0\right ) &= 6 \\ \end{align*}

[_separable]

3.477

3451

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

[_linear]

3.561

3452

\begin{align*} y^{\prime }+4 \tan \left (2 t \right ) y&=\tan \left (2 t \right ) \\ y \left (\frac {\pi }{8}\right ) &= 2 \\ \end{align*}

[_separable]

4.671

3453

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

[_linear]

2.397

3454

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

[_linear]

2.349

3455

\begin{align*} y^{\prime }&=-\cot \left (t \right ) y+6 \cos \left (t \right )^{2} \\ y \left (\frac {\pi }{4}\right ) &= 3 \\ \end{align*}

[_linear]

3.298

3456

\begin{align*} y^{\prime }-x y^{3}&=0 \\ \end{align*}

[_separable]

6.801

3457

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

[_separable]

4.672

3458

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

[_separable]

3.201

3459

\begin{align*} y \left (2 x^{2} y^{2}+1\right ) y^{\prime }+x \left (1+y^{4}\right )&=0 \\ \end{align*}

[_exact, _rational, [_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘]]

2.905

3460

\begin{align*} 2 x y^{\prime }+3 x +y&=0 \\ \end{align*}

[_linear]

11.503

3461

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

[[_1st_order, ‘_with_symmetry_[F(x)*G(y),0]‘], [_Abel, ‘2nd type‘, ‘class B‘]]

32.144

3462

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

[_linear]

8.500

3463

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

[_linear]

3.941

3464

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

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

5.137

3465

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

[_rational, _Bernoulli]

2.709

3466

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

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

12.445

3467

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

[[_homogeneous, ‘class C‘], [_Abel, ‘2nd type‘, ‘class C‘], _dAlembert]

4.693

3468

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

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

7.353

3469

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

[_separable]

4.948

3470

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

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

11.405

3471

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

[_linear]

8.081

3472

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

[_separable]

3.259

3473

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

[_linear]

2.883

3474

\begin{align*} y^{\prime }-y \tan \left (x \right )&=1 \\ y \left (\frac {\pi }{4}\right ) &= 3 \\ \end{align*}

[_linear]

2.403

3475

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

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

5.435

3476

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

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

4.554

3477

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

[_linear]

3.332

3478

\begin{align*} \left (5 x +y-7\right ) y^{\prime }&=3 x +3 y+3 \\ \end{align*}

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

28.006

3479

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

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

35.262

3480

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

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

8.968

3481

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

[[_1st_order, ‘_with_symmetry_[F(x),G(y)]‘]]

7.826

3482

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

[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

1.607

3483

\begin{align*} x^{\prime \prime }+\omega _{0}^{2} x&=a \cos \left (\omega t \right ) \\ x \left (0\right ) &= 0 \\ x^{\prime }\left (0\right ) &= 0 \\ \end{align*}

[[_2nd_order, _linear, _nonhomogeneous]]

0.996

3484

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

[[_2nd_order, _missing_x]]

0.575

3485

\begin{align*} f^{\prime \prime }+2 f^{\prime }+5 f&={\mathrm e}^{-t} \cos \left (3 t \right ) \\ f \left (0\right ) &= 0 \\ f^{\prime }\left (0\right ) &= 0 \\ \end{align*}

[[_2nd_order, _linear, _nonhomogeneous]]

0.711

3486

\begin{align*} f^{\prime \prime }+6 f^{\prime }+9 f&={\mathrm e}^{-t} \\ f \left (0\right ) &= 0 \\ f^{\prime }\left (0\right ) &= \lambda \\ \end{align*}

[[_2nd_order, _with_linear_symmetries]]

0.634

3487

\begin{align*} f^{\prime \prime }+8 f^{\prime }+12 f&=12 \,{\mathrm e}^{-4 t} \\ f \left (0\right ) &= 0 \\ f^{\prime }\left (0\right ) &= 0 \\ \end{align*}

[[_2nd_order, _with_linear_symmetries]]

0.613

3488

\begin{align*} f^{\prime \prime }+8 f^{\prime }+12 f&=12 \,{\mathrm e}^{-4 t} \\ f \left (0\right ) &= 0 \\ f^{\prime }\left (0\right ) &= -2 \\ \end{align*}

[[_2nd_order, _with_linear_symmetries]]

0.606

3489

\begin{align*} y+2 y^{\prime }+y^{\prime \prime }&=4 \,{\mathrm e}^{-x} \\ \end{align*}

[[_2nd_order, _with_linear_symmetries]]

0.563

3490

\begin{align*} y^{\prime \prime \prime }-12 y^{\prime }+16 y&=32 x -8 \\ \end{align*}

[[_3rd_order, _with_linear_symmetries]]

0.127

3491

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

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

5.196

3492

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

[[_2nd_order, _with_linear_symmetries]]

4.734

3493

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

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

2.092

3494

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

[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, ‘_with_symmetry_[0,F(x)]‘]]

0.471

3495

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.740

3496

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

[[_2nd_order, _linear, _nonhomogeneous]]

0.573

3497

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

[[_3rd_order, _exact, _nonlinear]]

0.052

3498

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

[[_3rd_order, _missing_y]]

0.250

3499

\begin{align*} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+6\right ) y&={\mathrm e}^{-x^{2}} \sin \left (2 x \right ) \\ \end{align*}

[[_2nd_order, _linear, _nonhomogeneous]]

1.011

3500

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

Series expansion around \(z=0\).

[_Gegenbauer]

0.572