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2.20.35 Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999

Column notations: A is ODE degree. B is Program Number of solutions generated. C is CAS Number of solutions generated.

Table 2.448: Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999

#

ODE

A

B

C

Program classification

CAS classification

Solved?

Verified?

time (sec)

5480

\[ {}\left (-1+x \right ) \left (-2+x \right ) y^{\prime \prime }+\left (4 x -6\right ) y^{\prime }+2 y = 0 \]

i.c.

1

2

1

second order series method. Ordinary point, second order series method. Taylor series method

[[_2nd_order, _exact, _linear, _homogeneous]]

✓

✓

1.69

5481

\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \]

i.c.

1

2

1

second order series method. Ordinary point, second order series method. Taylor series method

[[_2nd_order, _with_linear_symmetries]]

✓

✓

1.019

5482

\[ {}y^{\prime \prime }-2 x y^{\prime }+8 y = 0 \]

i.c.

1

2

1

second order series method. Ordinary point, second order series method. Taylor series method

[[_2nd_order, _with_linear_symmetries]]

✓

✓

0.874

5483

\[ {}\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y = 0 \]

i.c.

1

2

1

second order series method. Ordinary point, second order series method. Taylor series method

[_Gegenbauer]

✓

✓

1.562

5484

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

i.c.

1

2

1

second order series method. Ordinary point, second order series method. Taylor series method

[[_2nd_order, _with_linear_symmetries]]

✓

✓

1.369

5485

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

1

1

1

second order series method. Regular singular point. Repeated root

[[_2nd_order, _with_linear_symmetries]]

✓

✓

0.984

5486

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

1

1

1

second order series method. Regular singular point. Difference is integer

[[_Emden, _Fowler]]

✓

✓

2.671

5487

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

1

2

1

second order series method. Ordinary point, second order series method. Taylor series method

[[_2nd_order, _with_linear_symmetries]]

✓

✓

0.612

5488

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

1

1

1

second order series method. Regular singular point. Difference is integer

[[_2nd_order, _exact, _linear, _homogeneous]]

✓

✓

3.03

5489

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

1

1

1

second order series method. Regular singular point. Difference not integer

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

✓

✓

0.816

5490

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

1

1

1

second order series method. Regular singular point. Difference is integer

[[_2nd_order, _with_linear_symmetries]]

✓

✓

1.325

5491

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

1

2

1

second order series method. Ordinary point, second order series method. Taylor series method

[[_Emden, _Fowler]]

✓

✓

0.468

5492

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

1

1

1

second order series method. Regular singular point. Difference not integer

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

✓

✓

1.049

5493

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

1

1

1

second order series method. Regular singular point. Difference not integer

[[_2nd_order, _exact, _linear, _homogeneous]]

✓

✓

0.929

5494

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

1

1

1

second order series method. Regular singular point. Complex roots

[[_2nd_order, _with_linear_symmetries]]

✓

✓

1.945

5495

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}+\frac {9}{4}\right ) y = 0 \]

1

1

1

second order series method. Regular singular point. Complex roots

[[_2nd_order, _with_linear_symmetries]]

✓

✓

1.938

5496

\[ {}x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}+\frac {25}{4}\right ) y = 0 \]

1

1

1

second order series method. Regular singular point. Complex roots

[[_2nd_order, _with_linear_symmetries]]

✓

✓

2.287

5497

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

1

2

1

second order series method. Ordinary point, second order series method. Taylor series method

[[_2nd_order, _with_linear_symmetries]]

✓

✓

0.622

5498

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

1

2

1

first order ode series method. Ordinary point, first order ode series method. Taylor series method

[_linear]

✓

✓

0.548

5499

\[ {}y^{\prime }+x y = \frac {1}{x^{3}} \]

1

1

1

linear, exactWithIntegrationFactor, first_order_ode_lie_symmetry_lookup

[_linear]

✓

✓

0.727

5500

\[ {}x^{3} y^{\prime \prime }+y = \frac {1}{x^{4}} \]

1

0

0

second order series method. Irregular singular point

[[_2nd_order, _linear, _nonhomogeneous]]

❇

N/A

0.241

5501

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

1

0

0

second order series method. Regular singular point. Difference is integer

[[_2nd_order, _linear, _nonhomogeneous]]

❇

N/A

3.997

5502

\[ {}y^{\prime }-\frac {y}{x} = \cos \left (x \right ) \]

1

0

0

[_linear]

❇

N/A

0.401

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