2.2.84 Problems 8301 to 8400

Table 2.169: Main lookup table. Sorted sequentially by problem number.

#

ODE

CAS classification

Solved?

time (sec)

8301

\[ {}x^{2} y^{\prime \prime }+\left (x^{2}-2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.494

8302

\[ {}4 x^{2} y^{\prime \prime }+\left (16 x^{2}+1\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.743

8303

\[ {}x y^{\prime \prime }+3 y^{\prime }+x^{3} y = 0 \]

[[_Emden, _Fowler]]

1.503

8304

\[ {}9 x^{2} y^{\prime \prime }+9 x y^{\prime }+\left (x^{6}-36\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.824

8305

\[ {}y^{\prime \prime }-x^{2} y = 0 \]

[[_Emden, _Fowler]]

0.801

8306

\[ {}x y^{\prime \prime }+y^{\prime }-7 x^{3} y = 0 \]

[[_Emden, _Fowler]]

0.890

8307

\[ {}y^{\prime \prime }+y = 0 \]

[[_2nd_order, _missing_x]]

2.087

8308

\[ {}x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}+2\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.460

8309

\[ {}16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.581

8310

\[ {}4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (16 x^{2}+3\right ) y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

2.793

8311

\[ {}2 x y^{\prime \prime }+y^{\prime }+y = 0 \]

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

0.843

8312

\[ {}y^{\prime \prime }-x y^{\prime }-y = 0 \]

[[_2nd_order, _exact, _linear, _homogeneous]]

0.400

8313

\[ {}\left (-1+x \right ) y^{\prime \prime }+3 y = 0 \]

[[_Emden, _Fowler]]

0.493

8314

\[ {}y^{\prime \prime }-x^{2} y^{\prime }+x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

0.378

8315

\[ {}x y^{\prime \prime }-\left (x +2\right ) y^{\prime }+2 y = 0 \]

[_Laguerre]

0.793

8316

\[ {}\cos \left (x \right ) y^{\prime \prime }+y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.106

8317

\[ {}y^{\prime \prime }+x y^{\prime }+2 y = 0 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

0.390

8318

\[ {}\left (x +2\right ) y^{\prime \prime }+3 y = 0 \]
i.c.

[[_Emden, _Fowler]]

0.445

8319

\[ {}\left (1-2 \sin \left (x \right )\right ) y^{\prime \prime }+x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

4.782

8320

\[ {}y^{\prime \prime }+x y^{\prime }+y = 0 \]
i.c.

[[_2nd_order, _exact, _linear, _homogeneous]]

0.467

8321

\[ {}x y^{\prime \prime }+\left (-\cos \left (x \right )+1\right ) y^{\prime }+x^{2} y = 0 \]

[[_Emden, _Fowler]]

3.460

8322

\[ {}\left ({\mathrm e}^{x}-1-x \right ) y^{\prime \prime }+x y = 0 \]

[[_2nd_order, _with_linear_symmetries]]

1.995

8323

\[ {}y^{\prime \prime }+x^{2} y^{\prime }+2 x y = 10 x^{3}-2 x +5 \]

[[_2nd_order, _exact, _linear, _nonhomogeneous]]

0.504

8324

\[ {}y^{\prime }-y = 1 \]
i.c.

[_quadrature]

0.360

8325

\[ {}2 y^{\prime }+y = 0 \]
i.c.

[_quadrature]

0.352

8326

\[ {}y^{\prime }+6 y = {\mathrm e}^{4 t} \]
i.c.

[[_linear, ‘class A‘]]

0.402

8327

\[ {}y^{\prime }-y = 2 \cos \left (5 t \right ) \]
i.c.

[[_linear, ‘class A‘]]

0.591

8328

\[ {}y^{\prime \prime }+5 y^{\prime }+4 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

0.302

8329

\[ {}y^{\prime \prime }-4 y^{\prime } = 6 \,{\mathrm e}^{3 t}-3 \,{\mathrm e}^{-t} \]
i.c.

[[_2nd_order, _missing_y]]

0.351

8330

\[ {}y^{\prime \prime }+y = \sqrt {2}\, \sin \left (\sqrt {2}\, t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.470

8331

\[ {}y^{\prime \prime }+9 y = {\mathrm e}^{t} \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

0.379

8332

\[ {}2 y^{\prime \prime \prime }+3 y^{\prime \prime }-3 y^{\prime }-2 y = {\mathrm e}^{-t} \]
i.c.

[[_3rd_order, _with_linear_symmetries]]

0.395

8333

\[ {}y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y = \sin \left (3 t \right ) \]
i.c.

[[_3rd_order, _linear, _nonhomogeneous]]

0.459

8334

\[ {}y^{\prime }+y = {\mathrm e}^{-3 t} \cos \left (2 t \right ) \]
i.c.

[[_linear, ‘class A‘]]

0.569

8335

\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

0.371

8336

\[ {}y^{\prime }+4 y = {\mathrm e}^{-4 t} \]
i.c.

[[_linear, ‘class A‘]]

0.454

8337

\[ {}y^{\prime }-y = 1+t \,{\mathrm e}^{t} \]
i.c.

[[_linear, ‘class A‘]]

0.463

8338

\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

0.222

8339

\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = t^{3} {\mathrm e}^{2 t} \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.229

8340

\[ {}y^{\prime \prime }-6 y^{\prime }+9 y = t \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

0.324

8341

\[ {}y^{\prime \prime }-4 y^{\prime }+4 y = t^{3} \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.346

8342

\[ {}y^{\prime \prime }-6 y^{\prime }+13 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

0.355

8343

\[ {}2 y^{\prime \prime }+20 y^{\prime }+51 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

0.454

8344

\[ {}y^{\prime \prime }-y = {\mathrm e}^{t} \cos \left (t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.468

8345

\[ {}y^{\prime \prime }-2 y^{\prime }+5 y = t +1 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

0.416

8346

\[ {}y^{\prime \prime }+2 y^{\prime }+y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

0.274

8347

\[ {}y^{\prime \prime }+8 y^{\prime }+20 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

0.178

8348

\[ {}y^{\prime }+y = \left \{\begin {array}{cc} 0 & 0\le t <1 \\ 5 & 1\le t \end {array}\right . \]
i.c.

[[_linear, ‘class A‘]]

0.626

8349

\[ {}y^{\prime }+y = \left \{\begin {array}{cc} 1 & 0\le t <1 \\ -1 & 1\le t \end {array}\right . \]
i.c.

[[_linear, ‘class A‘]]

0.775

8350

\[ {}y^{\prime }+y = \left \{\begin {array}{cc} t & 0\le t <1 \\ 0 & 1\le t \end {array}\right . \]
i.c.

[[_linear, ‘class A‘]]

0.742

8351

\[ {}y^{\prime \prime }+4 y = \left \{\begin {array}{cc} 1 & 0\le t <1 \\ 0 & 1\le t \end {array}\right . \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

1.066

8352

\[ {}y^{\prime \prime }+4 y = \sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.851

8353

\[ {}y^{\prime \prime }-5 y^{\prime }+6 y = \operatorname {Heaviside}\left (t -1\right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

1.006

8354

\[ {}y^{\prime \prime }+y = \left \{\begin {array}{cc} 0 & 0\le t <\pi \\ 1 & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right . \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

1.094

8355

\[ {}y^{\prime \prime }+4 y^{\prime }+3 y = 1-\operatorname {Heaviside}\left (t -2\right )-\operatorname {Heaviside}\left (-4+t \right )+\operatorname {Heaviside}\left (t -6\right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

2.095

8356

\[ {}y^{\prime }+y = t \sin \left (t \right ) \]
i.c.

[[_linear, ‘class A‘]]

0.542

8357

\[ {}y^{\prime }-y = t \,{\mathrm e}^{t} \sin \left (t \right ) \]
i.c.

[[_linear, ‘class A‘]]

0.554

8358

\[ {}y^{\prime \prime }+9 y = \cos \left (3 t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.361

8359

\[ {}y^{\prime \prime }+y = \sin \left (t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.381

8360

\[ {}y^{\prime \prime }+16 y = \left \{\begin {array}{cc} \cos \left (4 t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right . \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.950

8361

\[ {}y^{\prime \prime }+y = \left \{\begin {array}{cc} 1 & 0\le t <\frac {\pi }{2} \\ \sin \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right . \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.866

8362

\[ {}t y^{\prime \prime }-y^{\prime } = 2 t^{2} \]
i.c.

[[_2nd_order, _missing_y]]

1.178

8363

\[ {}2 y^{\prime \prime }+t y^{\prime }-2 y = 10 \]
i.c.

[[_2nd_order, _with_linear_symmetries]]

0.946

8364

\[ {}y^{\prime \prime }+y = \sin \left (t \right )+t \sin \left (t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.423

8365

\[ {}y^{\prime }-3 y = \delta \left (t -2\right ) \]
i.c.

[[_linear, ‘class A‘]]

0.512

8366

\[ {}y^{\prime }+y = \delta \left (t -1\right ) \]
i.c.

[[_linear, ‘class A‘]]

0.614

8367

\[ {}y^{\prime \prime }+y = \delta \left (t -2 \pi \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.604

8368

\[ {}y^{\prime \prime }+16 y = \delta \left (t -2 \pi \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.603

8369

\[ {}y^{\prime \prime }+y = \delta \left (t -\frac {\pi }{2}\right )+\delta \left (t -\frac {3 \pi }{2}\right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.820

8370

\[ {}y^{\prime \prime }+y = \delta \left (t -2 \pi \right )+\delta \left (t -4 \pi \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.872

8371

\[ {}y^{\prime \prime }+2 y^{\prime } = \delta \left (t -1\right ) \]
i.c.

[[_2nd_order, _missing_y]]

0.997

8372

\[ {}y^{\prime \prime }-2 y^{\prime } = 1+\delta \left (t -2\right ) \]
i.c.

[[_2nd_order, _missing_y]]

0.901

8373

\[ {}y^{\prime \prime }+4 y^{\prime }+5 y = \delta \left (t -2 \pi \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.823

8374

\[ {}y^{\prime \prime }+2 y^{\prime }+y = \delta \left (t -1\right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

0.584

8375

\[ {}y^{\prime \prime }+4 y^{\prime }+13 y = \delta \left (t -\pi \right )+\delta \left (t -3 \pi \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

2.592

8376

\[ {}y^{\prime \prime }-7 y^{\prime }+6 y = {\mathrm e}^{t}+\delta \left (t -2\right )+\delta \left (-4+t \right ) \]
i.c.

[[_2nd_order, _linear, _nonhomogeneous]]

1.428

8377

\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = 0 \]
i.c.

[[_2nd_order, _missing_x]]

0.352

8378

\[ {}y^{\prime \prime }+2 y^{\prime }+10 y = \delta \left (t \right ) \]

[[_2nd_order, _linear, _nonhomogeneous]]

0.299

8379

\[ {}\left [\begin {array}{c} x^{\prime }=3 x-5 y \\ y^{\prime }=4 x+8 y \end {array}\right ] \]

system_of_ODEs

0.842

8380

\[ {}\left [\begin {array}{c} x^{\prime }=4 x-7 y \\ y^{\prime }=5 x \end {array}\right ] \]

system_of_ODEs

0.824

8381

\[ {}\left [\begin {array}{c} x^{\prime }=-3 x+4 y-9 z \\ y^{\prime }=6 x-y \\ z^{\prime }=10 x+4 y+3 z \end {array}\right ] \]

system_of_ODEs

10.450

8382

\[ {}\left [\begin {array}{c} x^{\prime }=x-y \\ y^{\prime }=x+2 z \\ z^{\prime }=z-x \end {array}\right ] \]

system_of_ODEs

8.547

8383

\[ {}\left [\begin {array}{c} x^{\prime }=x-y+z+t -1 \\ y^{\prime }=2 x+y-z-3 t^{2} \\ z^{\prime }=x+y+z+t^{2}-t +2 \end {array}\right ] \]

system_of_ODEs

3.882

8384

\[ {}\left [\begin {array}{c} x^{\prime }=-3 x+4 y+{\mathrm e}^{-t} \sin \left (2 t \right ) \\ y^{\prime }=5 x+9 z+4 \,{\mathrm e}^{-t} \cos \left (2 t \right ) \\ z^{\prime }=y+6 z-{\mathrm e}^{-t} \end {array}\right ] \]

system_of_ODEs

221.326

8385

\[ {}\left [\begin {array}{c} x^{\prime }=4 x+2 y+{\mathrm e}^{t} \\ y^{\prime }=-x+3 y-{\mathrm e}^{t} \end {array}\right ] \]

system_of_ODEs

1.207

8386

\[ {}\left [\begin {array}{c} x^{\prime }=7 x+5 y-9 z-8 \,{\mathrm e}^{-2 t} \\ y^{\prime }=4 x+y+z+2 \,{\mathrm e}^{5 t} \\ z^{\prime }=-2 y+3 z+{\mathrm e}^{5 t}-3 \,{\mathrm e}^{-2 t} \end {array}\right ] \]

system_of_ODEs

37.659

8387

\[ {}\left [\begin {array}{c} x^{\prime }=x-y+2 z+{\mathrm e}^{-t}-3 t \\ y^{\prime }=3 x-4 y+z+2 \,{\mathrm e}^{-t}+t \\ z^{\prime }=-2 x+5 y+6 z+2 \,{\mathrm e}^{-t}-t \end {array}\right ] \]

system_of_ODEs

194.127

8388

\[ {}\left [\begin {array}{c} x^{\prime }=3 x-7 y+4 \sin \left (t \right )+\left (-4+t \right ) {\mathrm e}^{4 t} \\ y^{\prime }=x+y+8 \sin \left (t \right )+\left (2 t +1\right ) {\mathrm e}^{4 t} \end {array}\right ] \]

system_of_ODEs

4.744

8389

\[ {}\left [\begin {array}{c} x^{\prime }=3 x-4 y \\ y^{\prime }=4 x-7 y \end {array}\right ] \]

system_of_ODEs

0.459

8390

\[ {}\left [\begin {array}{c} x^{\prime }=-2 x+5 y \\ y^{\prime }=-2 x+4 y \end {array}\right ] \]

system_of_ODEs

0.537

8391

\[ {}\left [\begin {array}{c} x^{\prime }=-x+\frac {y}{4} \\ y^{\prime }=x-y \end {array}\right ] \]

system_of_ODEs

0.453

8392

\[ {}\left [\begin {array}{c} x^{\prime }=2 x+y \\ y^{\prime }=-x \end {array}\right ] \]

system_of_ODEs

0.416

8393

\[ {}\left [\begin {array}{c} x^{\prime }=x+2 y+z \\ y^{\prime }=6 x-y \\ z^{\prime }=-x-2 y-z \end {array}\right ] \]

system_of_ODEs

0.526

8394

\[ {}\left [\begin {array}{c} x^{\prime }=x+z \\ y^{\prime }=x+y \\ z^{\prime }=-2 x-z \end {array}\right ] \]

system_of_ODEs

0.551

8395

\[ {}\left [\begin {array}{c} x^{\prime }=x+2 y \\ y^{\prime }=4 x+3 y \end {array}\right ] \]

system_of_ODEs

0.452

8396

\[ {}\left [\begin {array}{c} x^{\prime }=2 x+2 y \\ y^{\prime }=x+3 y \end {array}\right ] \]

system_of_ODEs

0.448

8397

\[ {}\left [\begin {array}{c} x^{\prime }=-4 x+2 y \\ y^{\prime }=-\frac {5 x}{2}+2 y \end {array}\right ] \]

system_of_ODEs

0.462

8398

\[ {}\left [\begin {array}{c} x^{\prime }=-\frac {5 x}{2}+2 y \\ y^{\prime }=\frac {3 x}{4}-2 y \end {array}\right ] \]

system_of_ODEs

0.468

8399

\[ {}\left [\begin {array}{c} x^{\prime }=10 x-5 y \\ y^{\prime }=8 x-12 y \end {array}\right ] \]

system_of_ODEs

0.470

8400

\[ {}\left [\begin {array}{c} x^{\prime }=-6 x+2 y \\ y^{\prime }=-3 x+y \end {array}\right ] \]

system_of_ODEs

0.437