2.17.76 Problems 7501 to 7600

Problem 7501

ODE	\[ \boxed {\left (x^{2}+6 x \right ) y^{\prime \prime }+\left (9+3 x \right ) y^{\prime }-3 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x +3\right ) x^{\frac {3}{4}} \left (x +6\right )^{\frac {3}{4}}}{\left (x \left (x +6\right )\right )^{\frac {3}{4}}}-\frac {2 c_{2} \left (x +6\right )^{\frac {1}{4}} \left (x^{2}+6 x +\frac {9}{2}\right ) x^{\frac {1}{4}}}{81 \left (x \left (x +6\right )\right )^{\frac {3}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x +3\right )+\frac {c_{2} \left (2 x^{2}+12 x +9\right )}{\sqrt {x^{2}+6 x}} \]

Problem 7502

ODE	\[ \boxed {t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+y t^{2}=0} \]
program solution	\[ y = c_{1} \left (t -1\right ) {\mathrm e}^{-\frac {t \left (t -2\right )}{2}}+c_{2} \left (t -1\right ) {\mathrm e}^{-\frac {t \left (t -2\right )}{2}} \left (\int \frac {t \,{\mathrm e}^{\frac {t \left (t -4\right )}{2}}}{\left (t -1\right )^{2}}d t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{t -\frac {1}{2} t^{2}} \left (t -1\right )+c_{2} {\mathrm e}^{t -\frac {1}{2} t^{2}} \left (t -1\right ) \left (\int \frac {t \,{\mathrm e}^{\frac {1}{2} t^{2}-2 t}}{\left (t -1\right )^{2}}d t \right ) \]

Problem 7503

ODE	\[ \boxed {t^{2} y^{\prime \prime }-t \left (t +2\right ) y^{\prime }+\left (t +2\right ) y=0} \]
program solution	\[ y = c_{1} t +c_{2} {\mathrm e}^{t} t \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} t +c_{2} t \,{\mathrm e}^{t} \]

Problem 7504

ODE	\[ \boxed {t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}+\left (-t -1\right ) c_{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \left (t +1\right )+c_{2} {\mathrm e}^{t} \]

Problem 7505

ODE	\[ \boxed {\left (-t +1\right ) y^{\prime \prime }+t y^{\prime }-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}-c_{2} t \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} t +c_{2} {\mathrm e}^{t} \]

Problem 7506

ODE	\[ \boxed {y^{\prime \prime } x^{2}+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \cos \left (x \right )}{\sqrt {x}}+\frac {c_{2} \sin \left (x \right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (x \right )}{\sqrt {x}}+\frac {c_{2} \cos \left (x \right )}{\sqrt {x}} \]

Problem 7507

ODE	\[ \boxed {t y^{\prime \prime }-\left (t +1\right ) y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}+\left (-t -1\right ) c_{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \left (t +1\right )+c_{2} {\mathrm e}^{t} \]

Problem 7508

ODE	\[ \boxed {\left (-t +1\right ) y^{\prime \prime }+t y^{\prime }-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}-c_{2} t \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} t +c_{2} {\mathrm e}^{t} \]

Problem 7509

ODE	\[ \boxed {y^{\prime \prime }+x y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} \left (-\frac {i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \,{\mathrm e}^{-\frac {x^{2}}{2}}}{2}-1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+\frac {c_{2} \sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}} \left (i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {x^{2}}{2}}-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \right )}{2 \sqrt {\pi }} \]

Problem 7510

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}+1\right )^{3}}{\left (i x +1\right )^{3}}+\frac {c_{2} \left (x^{2}+1\right )^{3} \left (x^{2}-\frac {1}{3}\right )}{\left (i x +1\right )^{3} \left (x +i\right )^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (-3 x^{2}+1\right )+c_{2} \left (x^{3}-3 x \right ) \]

Problem 7511

ODE	\[ \boxed {\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y=0} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} -x c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +{\mathrm e}^{x} c_{2} \]

Problem 7512

ODE	\[ \boxed {2 y^{\prime \prime }+x y^{\prime }+3 y=0} \]
program solution	\[ y = c_{1} \left (x^{2}-2\right ) {\mathrm e}^{-\frac {x^{2}}{4}}+c_{2} \left (x^{2}-2\right ) {\mathrm e}^{-\frac {x^{2}}{4}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{\left (x^{2}-2\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{2}}{4}} \left (x^{2}-2\right )+c_{2} {\mathrm e}^{-\frac {x^{2}}{4}} \left (x^{2}-2\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{\left (x^{2}-2\right )^{2}}d x \right ) \]

Problem 7513

ODE	\[ \boxed {y^{\prime \prime }+x y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} \left (-\frac {i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \,{\mathrm e}^{-\frac {x^{2}}{2}}}{2}-1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+\frac {c_{2} \sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}} \left (i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {x^{2}}{2}}-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \right )}{2 \sqrt {\pi }} \]

Problem 7514

ODE	\[ \boxed {\left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y=0} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} -x c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +{\mathrm e}^{x} c_{2} \]

Problem 7515

ODE	\[ \boxed {y^{\prime \prime }+x y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} \left (-\frac {i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \,{\mathrm e}^{-\frac {x^{2}}{2}}}{2}-1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+\frac {c_{2} \sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}} \left (i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {x^{2}}{2}}-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \right )}{2 \sqrt {\pi }} \]

Problem 7516

ODE	\[ \boxed {\left (-x^{2}+4\right ) y^{\prime \prime }+x y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} \sqrt {x^{2}-6}\, \left (x +\sqrt {x^{2}-4}\right )^{\sqrt {3}} {\mathrm e}^{-\frac {\operatorname {arctanh}\left (\frac {\sqrt {6}\, x -4}{\sqrt {2 x^{2}-8}}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {4+\sqrt {6}\, x}{\sqrt {2 x^{2}-8}}\right )}{2}}+c_{2} \sqrt {x^{2}-6}\, \left (x +\sqrt {x^{2}-4}\right )^{\sqrt {3}} {\mathrm e}^{-\frac {\operatorname {arctanh}\left (\frac {\sqrt {6}\, x -4}{\sqrt {2 x^{2}-8}}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {4+\sqrt {6}\, x}{\sqrt {2 x^{2}-8}}\right )}{2}} \left (\int \frac {\sqrt {x^{2}-4}\, \left (x +\sqrt {x^{2}-4}\right )^{-2 \sqrt {3}} {\mathrm e}^{\operatorname {arctanh}\left (\frac {\sqrt {6}\, x -4}{\sqrt {2 x^{2}-8}}\right )+\operatorname {arctanh}\left (\frac {4+\sqrt {6}\, x}{\sqrt {2 x^{2}-8}}\right )}}{x^{2}-6}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x^{2}-6}\, \sin \left (\int \frac {\sqrt {-x^{2}+4}\, \sqrt {3}}{x^{2}-6}d x \right )+c_{2} \sqrt {x^{2}-6}\, \cos \left (\int \frac {\sqrt {-x^{2}+4}\, \sqrt {3}}{x^{2}-6}d x \right ) \]

Problem 7517

ODE	\[ \boxed {4 y^{\prime \prime } x^{2}-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, {\mathrm e}^{-2 x}+\frac {c_{2} \sqrt {x}\, {\mathrm e}^{2 x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \sinh \left (2 x \right )+c_{2} \sqrt {x}\, \cosh \left (2 x \right ) \]

Problem 7518

ODE	\[ \boxed {\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y=0} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} -x c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +{\mathrm e}^{x} c_{2} \]

Problem 7519

ODE	\[ \boxed {y^{\prime \prime } x^{2}-2 x y^{\prime }+\left (x^{2}+2\right ) y=0} \]
program solution	\[ y = c_{1} \cos \left (x \right ) x +c_{2} x \sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \sin \left (x \right )+c_{2} \cos \left (x \right ) x \]

Problem 7520

ODE	\[ \boxed {\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{x} \sqrt {x}\, \sqrt {x -2}}{\sqrt {x \left (x -2\right )}}-\frac {c_{2} x^{\frac {5}{2}} \sqrt {x -2}}{\sqrt {x \left (x -2\right )}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2}+{\mathrm e}^{x} c_{2} \]

Problem 7521

ODE	\[ \boxed {4 y^{\prime \prime } x^{2}+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{x}}{\sqrt {x}}+\sqrt {x}\, c_{2} {\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{x}}{\sqrt {x}}+c_{2} \sqrt {x}\, {\mathrm e}^{x} \]

Problem 7522

ODE	\[ \boxed {y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x^{2}}+c_{2} x \,{\mathrm e}^{-x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{2}}+c_{2} {\mathrm e}^{-x^{2}} x \]

Problem 7523

ODE	\[ \boxed {\left (1+2 x \right ) y^{\prime \prime }-2 y^{\prime }-\left (3+2 x \right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {2}\, {\mathrm e}^{-x}+\frac {c_{2} x \sqrt {2}\, {\mathrm e}^{x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +c_{2} x \,{\mathrm e}^{x} \]

Problem 7524

ODE	\[ \boxed {x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (x +2\right ) y=0} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} +\frac {c_{2} x^{3} {\mathrm e}^{x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{x} x^{3} \]

Problem 7525

ODE	\[ \boxed {y^{\prime \prime } x^{2}-2 x y^{\prime }+\left (x^{2}+2\right ) y=0} \]
program solution	\[ y = c_{1} \cos \left (x \right ) x +c_{2} x \sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \sin \left (x \right )+c_{2} \cos \left (x \right ) x \]

Problem 7526

ODE	\[ \boxed {4 y^{\prime \prime } x^{2}-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, {\mathrm e}^{-2 x}+\frac {c_{2} \sqrt {x}\, {\mathrm e}^{2 x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \sinh \left (2 x \right )+c_{2} \sqrt {x}\, \cosh \left (2 x \right ) \]

Problem 7527

ODE	\[ \boxed {4 y^{\prime \prime } x^{2}-4 x y^{\prime }+\left (4 x^{2}+3\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \cos \left (x \right )+c_{2} \sqrt {x}\, \sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \sin \left (x \right )+c_{2} \sqrt {x}\, \cos \left (x \right ) \]

Problem 7528

ODE	\[ \boxed {y^{\prime \prime } x^{2}-2 x y^{\prime }-\left (x^{2}-2\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x} x +\frac {{\mathrm e}^{x} c_{2} x}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \sinh \left (x \right )+c_{2} x \cosh \left (x \right ) \]

Problem 7529

ODE	\[ \boxed {y^{\prime \prime } x^{2}-2 x \left (1+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y=0} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} x +c_{2} {\mathrm e}^{x} x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{x}+c_{2} {\mathrm e}^{x} x^{2} \]

Problem 7530

ODE	\[ \boxed {y^{\prime \prime } x^{2}-2 x \left (x +2\right ) y^{\prime }+\left (x^{2}+4 x +6\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} x^{2}+c_{2} x^{3} {\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{x} x^{2}+c_{2} {\mathrm e}^{x} x^{3} \]

Problem 7531

ODE	\[ \boxed {y^{\prime \prime } x^{2}-4 x y^{\prime }+\left (x^{2}+6\right ) y=0} \]
program solution	\[ y = \cos \left (x \right ) c_{1} x^{2}+x^{2} \sin \left (x \right ) c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \sin \left (x \right )+c_{2} \cos \left (x \right ) x^{2} \]

Problem 7532

ODE	\[ \boxed {\left (x -1\right ) y^{\prime \prime }-x y^{\prime }+y=0} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} -x c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +{\mathrm e}^{x} c_{2} \]

Problem 7533

ODE	\[ \boxed {4 y^{\prime \prime } x^{2}-4 x \left (1+x \right ) y^{\prime }+\left (3+2 x \right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}+\sqrt {x}\, c_{2} {\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}+c_{2} \sqrt {x}\, {\mathrm e}^{x} \]

Problem 7534

ODE	\[ \boxed {\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {3}\, {\mathrm e}^{2 x}-\frac {c_{2} x \sqrt {3}\, {\mathrm e}^{-x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{2 x}+{\mathrm e}^{-x} c_{2} x \]

Problem 7535

ODE	\[ \boxed {\left (x +2\right ) y^{\prime \prime }+x y^{\prime }+3 y=0} \]
program solution	\[ y = c_{1} \left (x^{2}-6 x +4\right ) \left (x +2\right )^{3} {\mathrm e}^{-x}+c_{2} \left (-\frac {\left (x^{2}-6 x +4\right ) \left (x +2\right )^{3} {\mathrm e}^{-2-x} \operatorname {expIntegral}_{1}\left (-2-x \right )}{240}-\frac {x^{4}}{240}+\frac {x^{3}}{240}+\frac {3 x^{2}}{40}+\frac {11 x}{120}-\frac {1}{30}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x} \left (x^{5}-20 x^{3}-40 x^{2}+32\right )-\frac {c_{2} \left ({\mathrm e}^{-2} \operatorname {expIntegral}_{1}\left (-2-x \right ) x^{5}+{\mathrm e}^{x} x^{4}-20 \,{\mathrm e}^{-2} \operatorname {expIntegral}_{1}\left (-2-x \right ) x^{3}-{\mathrm e}^{x} x^{3}-40 \,{\mathrm e}^{-2} \operatorname {expIntegral}_{1}\left (-2-x \right ) x^{2}-18 \,{\mathrm e}^{x} x^{2}-22 x \,{\mathrm e}^{x}+32 \,{\mathrm e}^{-2} \operatorname {expIntegral}_{1}\left (-2-x \right )+8 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{240} \]

Problem 7536

ODE	\[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (-x +2\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}+6 x +3\right )}{x}+\frac {c_{2} \left (1+3 \left (x^{3}+6 x^{2}+3 x \right ) \ln \left (x \right )+51 x^{2}+48 x \right )}{3 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+6 x +3\right )}{x}+\frac {c_{2} \left (3 \ln \left (x \right ) x^{3}+18 x^{2} \ln \left (x \right )+9 x \ln \left (x \right )+51 x^{2}+48 x +1\right )}{3 x^{2}} \]

Problem 7537

ODE	\[ \boxed {x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }-\left (6 x +4\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \sqrt {1+x}\, x^{\frac {5}{2}}}{\sqrt {x \left (1+x \right )}}+\frac {c_{2} \sqrt {1+x}\, \left (12 \ln \left (x \right ) x^{4}-12 \ln \left (1+x \right ) x^{4}+12 x^{3}-6 x^{2}+4 x -3\right )}{12 x^{\frac {3}{2}} \sqrt {x \left (1+x \right )}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2}+\frac {c_{2} \left (12 x^{4} \ln \left (x \right )-12 \ln \left (x +1\right ) x^{4}+12 x^{3}-6 x^{2}+4 x -3\right )}{12 x^{2}} \]

Problem 7538

ODE	\[ \boxed {x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \left (1-x^{2}\right ) y=0} \]
program solution	\[ y = \frac {c_{1} 2^{\frac {1}{4}}}{2 x}+\frac {c_{2} \left (\left (x^{2}-1\right ) \sqrt {2}\, \sqrt {2 x^{2}+1}+3 \,\operatorname {arcsinh}\left (\sqrt {2}\, x \right ) x \right ) 2^{\frac {1}{4}}}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{x}+\frac {c_{2} \sqrt {2}\, \left (\sqrt {2}\, \sqrt {2 x^{2}+1}\, x^{2}+3 \,\operatorname {arcsinh}\left (\sqrt {2}\, x \right ) x -\sqrt {2}\, \sqrt {2 x^{2}+1}\right )}{2 x^{2}} \]

Problem 7539

ODE	\[ \boxed {x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}+8\right )}{x}+\frac {c_{2} \left (-x^{2} \sqrt {2}\, \left (x^{2}+8\right ) \operatorname {arctanh}\left (\frac {\sqrt {2}}{\sqrt {x^{2}+2}}\right )+\left (2 x^{2}-8\right ) \sqrt {x^{2}+2}\right )}{64 x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+8\right )}{x}-\frac {c_{2} \sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {2}}{\sqrt {x^{2}+2}}\right ) x^{4}-\sqrt {2}\, \sqrt {x^{2}+2}\, x^{2}+8 \,\operatorname {arctanh}\left (\frac {\sqrt {2}}{\sqrt {x^{2}+2}}\right ) x^{2}+4 \sqrt {2}\, \sqrt {x^{2}+2}\right )}{64 x^{3}} \]

Problem 7540

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+6 y=0} \]
program solution	\[ y = \frac {c_{1}}{\left (i x +1\right )^{2}}+\frac {c_{2} x}{\left (-x +i\right )^{2} \left (x +i\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x}{\left (x^{2}+1\right )^{2}}+\frac {c_{2} \left (x^{2}-1\right )}{\left (x^{2}+1\right )^{2}} \]

Problem 7541

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }-2 y=0} \]
program solution	\[ y = c_{1} x +c_{2} \left (-\arctan \left (x \right ) x -1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +c_{2} \left (\arctan \left (x \right ) x +1\right ) \]

Problem 7542

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+20 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}+1\right )^{5}}{\left (i x +1\right )^{5}}+\frac {c_{2} \left (x^{2}+1\right )^{5} \left (x^{4}-2 x^{2}+\frac {1}{5}\right )}{\left (i x +1\right )^{5} \left (x +i\right )^{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (5 x^{4}-10 x^{2}+1\right )+c_{2} \left (x^{5}-10 x^{3}+5 x \right ) \]

Problem 7543

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }-12 y=0} \]
program solution	\[ y = \frac {c_{1}}{\left (x -1\right )^{3}}+\frac {c_{2} \left (-3 x^{2}-1\right )}{3 \left (x -1\right )^{3} \left (x +1\right )^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (3 x^{2}+1\right )}{\left (x -1\right )^{3} \left (x +1\right )^{3}}+\frac {c_{2} \left (x^{3}+3 x \right )}{\left (x -1\right )^{3} \left (x +1\right )^{3}} \]

Problem 7544

ODE	\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y=0} \]
program solution	\[ y = \frac {c_{1} x 2^{\frac {1}{8}}}{\left (2 x^{2}+1\right )^{\frac {3}{4}}}+\frac {c_{2} x 2^{\frac {7}{8}} \left (\int \frac {1}{\left (2 x^{2}+1\right )^{\frac {1}{4}} x^{2}}d x \right )}{2 \left (2 x^{2}+1\right )^{\frac {3}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x}{\left (2 x^{2}+1\right )^{\frac {3}{4}}}+\frac {c_{2} x \left (\int \frac {1}{\left (2 x^{2}+1\right )^{\frac {1}{4}} x^{2}}d x \right )}{\left (2 x^{2}+1\right )^{\frac {3}{4}}} \]

Problem 7545

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y=0} \]
program solution	\[ y = \frac {c_{1} x}{\left (x^{2}-1\right )^{\frac {1}{4}} \left (x -1\right )^{\frac {5}{4}} \left (x +1\right )^{\frac {5}{4}}}+\frac {c_{2} \left (\ln \left (x +\sqrt {x^{2}-1}\right ) x -\sqrt {x^{2}-1}\right )}{\left (x^{2}-1\right )^{\frac {1}{4}} \left (x -1\right )^{\frac {5}{4}} \left (x +1\right )^{\frac {5}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x}{\left (x^{2}-1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (\ln \left (x +\sqrt {x^{2}-1}\right ) x -\sqrt {x^{2}-1}\right )}{\left (x^{2}-1\right )^{\frac {3}{2}}} \]

Problem 7546

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }+28 y=0} \]
program solution	\[ y = \frac {c_{1} \left (-x +6 i\right ) \left (x^{2}+1\right )^{6}}{\left (-x +i\right )^{6}}+\frac {c_{2} \left (x^{2}+1\right )^{6} \left (35 x^{4}-42 x^{2}+3\right )}{105 \left (-x +i\right )^{6} \left (x +i\right )^{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {35}{3} x^{4}-14 x^{2}\right )+c_{2} \left (x^{7}+21 x^{5}-105 x^{3}+35 x \right ) \]

Problem 7547

ODE	\[ \boxed {y^{\prime \prime }+x y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} \left (-\frac {i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \,{\mathrm e}^{-\frac {x^{2}}{2}}}{2}-1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+\frac {c_{2} \sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}} \left (i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {x^{2}}{2}}-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x \right )}{2 \sqrt {\pi }} \]

Problem 7548

ODE	\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }-9 x y^{\prime }-6 y=0} \]
program solution	\[ y = \frac {c_{1} \left (3 x^{6}+5 x^{4}+3 x^{2}+1\right ) 2^{\frac {7}{8}}}{12}+c_{2} \left (18 x^{6}+30 x^{4}+18 x^{2}+6\right ) 2^{\frac {1}{8}} \left (\int \frac {\left (2 x^{2}+1\right )^{\frac {9}{4}}}{\left (3 x^{6}+5 x^{4}+3 x^{2}+1\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (3 x^{6}+5 x^{4}+3 x^{2}+1\right )+c_{2} \left (3 x^{6}+5 x^{4}+3 x^{2}+1\right ) \left (\int \frac {\left (2 x^{2}+1\right )^{\frac {9}{4}}}{\left (3 x^{4}+2 x^{2}+1\right )^{2} \left (x^{2}+1\right )^{2}}d x \right ) \]

Problem 7549

ODE	\[ \boxed {\left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (x -2\right ) y^{\prime }+36 y=0} \]
program solution	\[ y = -\frac {c_{1} \left (5 i \sqrt {6}-2 x +4\right ) \left (2 x -4+i \sqrt {6}\right )^{3} \left (2 x^{2}-8 x +11\right )^{2}}{4 \left (-2 x +4+i \sqrt {6}\right )^{2}}-\frac {8 c_{2} \left (2 x^{2}-8 x +11\right )^{2} \left (10 x^{3}-60 x^{2}+111 x -62\right )}{15 \left (2 x -4+i \sqrt {6}\right )^{2} \left (-2 x +4+i \sqrt {6}\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (-\frac {31}{5}+x^{3}-6 x^{2}+\frac {111}{10} x \right )+c_{2} \left (x^{6}-12 x^{5}+\frac {165}{2} x^{4}-\frac {16577}{8}-\frac {5445}{4} x^{2}+3267 x \right ) \]

Problem 7550

ODE	\[ \boxed {y^{\prime \prime }+\left (x -3\right ) y^{\prime }+3 y=0} \]
program solution	\[ y = c_{1} \left (x^{2}-6 x +8\right ) {\mathrm e}^{-\frac {x \left (x -6\right )}{2}}+c_{2} \left (x^{2}-6 x +8\right ) {\mathrm e}^{-\frac {x \left (x -6\right )}{2}} \left (\int \frac {{\mathrm e}^{\frac {x \left (x -6\right )}{2}}}{\left (x^{2}-6 x +8\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {1}{2} x^{2}+3 x} \left (x^{2}-6 x +8\right )+c_{2} {\mathrm e}^{-\frac {1}{2} x^{2}+3 x} \left (x^{2}-6 x +8\right ) \left (\int \frac {{\mathrm e}^{\frac {1}{2} x^{2}-3 x}}{\left (x -2\right )^{2} \left (x -4\right )^{2}}d x \right ) \]

Problem 7551

ODE	\[ \boxed {\left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x -4+\sqrt {2}\right )^{3} \left (x^{2}-8 x +14\right )^{2}}{\left (-x +4+\sqrt {2}\right )^{2}}-\frac {c_{2} \left (x^{4}-16 x^{3}+100 x^{2}-288 x +\frac {1604}{5}\right ) \left (x^{2}-8 x +14\right )^{2}}{\left (x -4-\sqrt {2}\right )^{2} \left (x -4+\sqrt {2}\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (\frac {1604}{5}+x^{4}-16 x^{3}+100 x^{2}-288 x \right )+c_{2} \left (x^{5}-140 x^{3}+1120 x^{2}-3500 x +4032\right ) \]

Problem 7552

ODE	\[ \boxed {\left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (x +1\right ) y^{\prime }+60 y=0} \]
program solution	\[ y = \frac {c_{1} \left (2 x +2+i \sqrt {6}\right )^{\frac {7}{2}} \left (2 x^{2}+4 x +5\right )^{\frac {5}{2}}}{2 \left (2 x +2-i \sqrt {6}\right )^{\frac {5}{2}}}+\frac {c_{2} \left (-32 x^{5}-160 x^{4}-160 x^{3}+160 x^{2}+248 x +56\right ) \left (2 x^{2}+4 x +5\right )^{\frac {5}{2}}}{\left (2 x +2+i \sqrt {6}\right )^{\frac {5}{2}} \left (2 x +2-i \sqrt {6}\right )^{\frac {5}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (-\frac {7}{4}+x^{5}+5 x^{4}+5 x^{3}-5 x^{2}-\frac {31}{4} x \right )+c_{2} \left (x^{6}+\frac {155}{8}-\frac {75}{2} x^{4}-100 x^{3}-\frac {225}{4} x^{2}+30 x \right ) \]

Problem 7553

ODE	\[ \boxed {\left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 y x=0} \]
program solution	\[ y = \frac {c_{1} x}{\left (x^{3}+1\right )^{\frac {4}{3}}}+\frac {c_{2} x \left (\int \frac {\left (x^{3}+1\right )^{\frac {1}{3}}}{x^{2}}d x \right )}{\left (x^{3}+1\right )^{\frac {4}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x}{\left (x^{3}+1\right )^{\frac {4}{3}}}+\frac {c_{2} x \left (\int \frac {{\left (\left (x +1\right ) \left (x^{2}-x +1\right )\right )}^{\frac {1}{3}}}{x^{2}}d x \right )}{\left (x^{3}+1\right )^{\frac {4}{3}}} \]

Problem 7554

ODE	\[ \boxed {\left (2 x^{5}+1\right ) y^{\prime \prime }+14 x^{4} y^{\prime }+10 x^{3} y=0} \]
program solution	\[ y = \frac {c_{1} x}{\left (2 x^{5}+1\right )^{\frac {2}{5}}}+\frac {c_{2} x \left (\int \frac {1}{\left (2 x^{5}+1\right )^{\frac {3}{5}} x^{2}}d x \right )}{\left (2 x^{5}+1\right )^{\frac {2}{5}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x}{\left (2 x^{5}+1\right )^{\frac {2}{5}}}+\frac {c_{2} x \left (\int \frac {1}{\left (2 x^{5}+1\right )^{\frac {3}{5}} x^{2}}d x \right )}{\left (2 x^{5}+1\right )^{\frac {2}{5}}} \]

Problem 7555

ODE	\[ \boxed {y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y=0} \]
program solution	\[ y = c_{1} x \,{\mathrm e}^{-\frac {x^{7}}{7}}+\frac {c_{2} \left (-7 \left (-x^{7}\right )^{\frac {6}{7}}+x^{7} 7^{\frac {6}{7}} {\mathrm e}^{-\frac {x^{7}}{7}} \left (\Gamma \left (\frac {6}{7}\right )-\Gamma \left (\frac {6}{7}, -\frac {x^{7}}{7}\right )\right )\right )}{7 \left (-x^{7}\right )^{\frac {6}{7}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{7}}{7}} x +\frac {7 c_{2} \left (-1\right )^{\frac {6}{7}} {\mathrm e}^{-\frac {x^{7}}{7}} \left (-\Gamma \left (\frac {6}{7}\right ) x^{7}+\left (-x^{7}\right )^{\frac {6}{7}} 7^{\frac {1}{7}} {\mathrm e}^{\frac {x^{7}}{7}}+\Gamma \left (\frac {6}{7}, -\frac {x^{7}}{7}\right ) x^{7}\right )}{\left (-x^{7}\right )^{\frac {6}{7}}} \]

Problem 7556

ODE	\[ \boxed {\left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y=0} \]
program solution	\[ y = \frac {16 c_{1} \left (\left (\left (i x^{6}+\frac {15}{7} x^{4}+\frac {15}{7} x^{2}-i\right ) \sqrt {2}+\left (1+i\right ) x^{6}+\left (\frac {15}{7}+\frac {15 i}{7}\right ) x^{4}+\left (\frac {15}{7}-\frac {15 i}{7}\right ) x^{2}+1-i\right ) \sqrt {2-\sqrt {2}}+\frac {12 x \left (\sqrt {2}\, \left (\left (1+i\right ) x^{4}+1-i\right )+\frac {8 x^{2}}{3}\right )}{7}\right ) \left (\left (-\sqrt {2}-1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right )^{2} \left (x^{8}+1\right )}{\left (\left (-i \sqrt {2}-1-i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \sqrt {2-\sqrt {2}}\, \left (\left (i \sqrt {2}+1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\left (i \sqrt {2}-1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\left (\sqrt {2}+1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\frac {\left (\sqrt {2}+1+i\right ) \sqrt {2-\sqrt {2}}}{2}+i x \right ) \left (\sqrt {2}+1-i\right ) \left (-\left (\sqrt {2}+1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\left (\sqrt {2}+1-i\right ) \sqrt {2-\sqrt {2}}+2 x \right )}+\frac {128 c_{2} \left (x^{8}-\frac {7}{9}\right ) \left (x^{8}+1\right ) \left (\left (\left (i x^{6}+\frac {15}{7} x^{4}+\frac {15}{7} x^{2}-i\right ) \sqrt {2}+\left (1+i\right ) x^{6}+\left (\frac {15}{7}+\frac {15 i}{7}\right ) x^{4}+\left (\frac {15}{7}-\frac {15 i}{7}\right ) x^{2}+1-i\right ) \sqrt {2-\sqrt {2}}+\frac {12 x \left (\sqrt {2}\, \left (\left (1+i\right ) x^{4}+1-i\right )+\frac {8 x^{2}}{3}\right )}{7}\right )}{\sqrt {2-\sqrt {2}}\, \left (\frac {6 \left (\left (i x^{4}+\frac {4}{3} x^{2}+1\right ) \sqrt {2}+\left (1+i\right ) x^{4}+\left (\frac {4}{3}+\frac {4 i}{3}\right ) x^{2}+1-i\right ) x \sqrt {2-\sqrt {2}}}{7}+\frac {\sqrt {2}\, \left (1-i+\left (\frac {15}{7}+\frac {15 i}{7}\right ) x^{4}\right )}{2}+i x^{6}+\frac {15 x^{2}}{7}\right ) \left (-\left (\sqrt {2}+1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\left (i \sqrt {2}-1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\left (\sqrt {2}+1-i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\frac {\left (\sqrt {2}+1+i\right ) \sqrt {2-\sqrt {2}}}{2}+i x \right ) \left (\left (i \sqrt {2}+1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\left (-\sqrt {2}-1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\left (\sqrt {2}+1+i\right ) \sqrt {2-\sqrt {2}}+2 x \right ) \left (\sqrt {2}+1-i\right ) \left (\left (\sqrt {2}+1-i\right ) \sqrt {2-\sqrt {2}}+2 i x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (-\frac {7}{9}+x^{8}\right )+c_{2} \left (x^{9}-\frac {9}{7} x \right ) \]

Problem 7557

ODE	\[ \boxed {y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y=0} \]
program solution	\[ y = c_{1} x \,{\mathrm e}^{-\frac {x^{6}}{6}}+\frac {c_{2} \left (-6 \left (-x^{6}\right )^{\frac {5}{6}}+x^{6} 6^{\frac {5}{6}} {\mathrm e}^{-\frac {x^{6}}{6}} \left (\Gamma \left (\frac {5}{6}\right )-\Gamma \left (\frac {5}{6}, -\frac {x^{6}}{6}\right )\right )\right )}{6 \left (-x^{6}\right )^{\frac {5}{6}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{6}}{6}} x -\frac {2 c_{2} {\mathrm e}^{-\frac {x^{6}}{6}} \left (\left (-x^{6}\right )^{\frac {5}{6}} 6^{\frac {2}{3}} \sqrt {3}\, \sqrt {2}\, {\mathrm e}^{\frac {x^{6}}{6}}-6 \Gamma \left (\frac {5}{6}\right ) x^{6}+6 \Gamma \left (\frac {5}{6}, -\frac {x^{6}}{6}\right ) x^{6}\right )}{\left (-x^{6}\right )^{\frac {5}{6}} \left (\sqrt {3}+i\right )} \]

Problem 7558

ODE	\[ \boxed {\left (1+3 x \right ) y^{\prime \prime }+x y^{\prime }+2 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x -6\right ) 3^{\frac {17}{18}} \left (1+3 x \right )^{\frac {10}{9}} {\mathrm e}^{-\frac {x}{3}}}{9}+3 c_{2} \left (x -6\right ) 3^{\frac {1}{18}} \left (1+3 x \right )^{\frac {10}{9}} {\mathrm e}^{-\frac {x}{3}} \left (\int \frac {{\mathrm e}^{\frac {x}{3}}}{\left (x -6\right )^{2} \left (1+3 x \right )^{\frac {19}{9}}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (3 x +1\right )^{\frac {10}{9}} {\mathrm e}^{-\frac {x}{3}} \left (x -6\right )+c_{2} \left (3 x +1\right )^{\frac {10}{9}} {\mathrm e}^{-\frac {x}{3}} \left (x -6\right ) \left (\int \frac {{\mathrm e}^{\frac {x}{3}}}{\left (x -6\right )^{2} \left (3 x +1\right )^{\frac {19}{9}}}d x \right ) \]

Problem 7559

ODE	\[ \boxed {\left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y=0} \]
program solution	\[ y = \frac {c_{1} x \sqrt {2}\, 3^{\frac {3}{4}} {\mathrm e}^{\frac {\sqrt {11}\, \arctan \left (\frac {\left (6 x +1\right ) \sqrt {11}}{11}\right )}{11}}}{6 \left (3 x^{2}+x +1\right )^{\frac {3}{2}}}+\frac {c_{2} x \sqrt {2}\, 3^{\frac {1}{4}} {\mathrm e}^{\frac {\sqrt {11}\, \arctan \left (\frac {\left (6 x +1\right ) \sqrt {11}}{11}\right )}{11}} \left (\int \frac {\sqrt {3 x^{2}+x +1}\, {\mathrm e}^{-\frac {\sqrt {11}\, \arctan \left (\frac {\left (6 x +1\right ) \sqrt {11}}{11}\right )}{11}}}{x^{2}}d x \right )}{\left (3 x^{2}+x +1\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (\frac {i \sqrt {11}-6 x -1}{i \sqrt {11}+6 x +1}\right )^{-\frac {i \sqrt {11}}{22}} x}{\left (3 x^{2}+x +1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (\frac {i \sqrt {11}-6 x -1}{i \sqrt {11}+6 x +1}\right )^{-\frac {i \sqrt {11}}{22}} x \left (\int \frac {\sqrt {3 x^{2}+x +1}\, \left (\frac {i \sqrt {11}+6 x +1}{i \sqrt {11}-6 x -1}\right )^{-\frac {i \sqrt {11}}{22}}}{x^{2}}d x \right )}{\left (3 x^{2}+x +1\right )^{\frac {3}{2}}} \]

Problem 7560

ODE	\[ \boxed {\left (2+x \right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }+3 y=0} \]
program solution	\[ y = c_{1} x \left (x -4\right ) \left (2+x \right )^{2} {\mathrm e}^{-x}+c_{2} \left (-\frac {x \left (x -4\right ) \left (2+x \right )^{2} {\mathrm e}^{-2-x} \operatorname {expIntegral}_{1}\left (-2-x \right )}{48}-\frac {x^{3}}{48}+\frac {x^{2}}{48}+\frac {5 x}{24}+\frac {1}{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x} x \left (x^{3}-12 x -16\right )-\frac {c_{2} \left ({\mathrm e}^{-2} \operatorname {expIntegral}_{1}\left (-2-x \right ) x^{4}+{\mathrm e}^{x} x^{3}-12 \,{\mathrm e}^{-2} \operatorname {expIntegral}_{1}\left (-2-x \right ) x^{2}-{\mathrm e}^{x} x^{2}-16 \,{\mathrm e}^{-2} \operatorname {expIntegral}_{1}\left (-2-x \right ) x -10 x \,{\mathrm e}^{x}-6 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{48} \]

Problem 7561

ODE	\[ \boxed {\left (4+x \right ) y^{\prime \prime }+\left (2+x \right ) y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} x \left (4+x \right )^{3} {\mathrm e}^{-x}+c_{2} \left (-\frac {x \left (4+x \right )^{3} {\mathrm e}^{-4-x} \operatorname {expIntegral}_{1}\left (-4-x \right )}{24}-\frac {x^{3}}{24}-\frac {3 x^{2}}{8}-\frac {11 x}{12}-\frac {1}{4}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x} x \left (x^{3}+12 x^{2}+48 x +64\right )-\frac {c_{2} \left ({\mathrm e}^{-4} \operatorname {expIntegral}_{1}\left (-x -4\right ) x^{4}+12 \,{\mathrm e}^{-4} \operatorname {expIntegral}_{1}\left (-x -4\right ) x^{3}+{\mathrm e}^{x} x^{3}+48 \,{\mathrm e}^{-4} \operatorname {expIntegral}_{1}\left (-x -4\right ) x^{2}+9 \,{\mathrm e}^{x} x^{2}+64 \,{\mathrm e}^{-4} \operatorname {expIntegral}_{1}\left (-x -4\right ) x +22 x \,{\mathrm e}^{x}+6 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{24} \]

Problem 7562

ODE	\[ \boxed {\left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (x +1\right ) y^{\prime }+8 y=0} \]
program solution	\[ y = \frac {c_{1} \left (2+x \right ) 2^{\frac {5}{6}}}{2 x^{\frac {7}{3}} \left (2 x +3\right )^{\frac {2}{3}}}+\frac {c_{2} \left (2+x \right ) 2^{\frac {1}{6}} \left (\int \frac {x^{\frac {4}{3}}}{\left (2 x +3\right )^{\frac {1}{3}} \left (2+x \right )^{2}}d x \right )}{x^{\frac {7}{3}} \left (2 x +3\right )^{\frac {2}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x +2\right )}{x^{\frac {7}{3}} \left (2 x +3\right )^{\frac {2}{3}}}+\frac {c_{2} \left (x +2\right ) \left (\int \frac {x^{\frac {4}{3}}}{\left (x +2\right )^{2} \left (2 x +3\right )^{\frac {1}{3}}}d x \right )}{x^{\frac {7}{3}} \left (2 x +3\right )^{\frac {2}{3}}} \]

Problem 7563

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x -2\right ) {\mathrm e}^{-\frac {6}{x}}}{x^{5}}+\frac {c_{2} \left (108 \left (x -2\right ) {\mathrm e}^{-\frac {6}{x}} \operatorname {expIntegral}_{1}\left (-\frac {6}{x}\right )+x^{3}+12 x^{2}-36 x \right )}{2 x^{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-\frac {6}{x}} \left (x -2\right )}{x^{5}}+\frac {c_{2} \left (x^{3} {\mathrm e}^{\frac {6}{x}}+12 x^{2} {\mathrm e}^{\frac {6}{x}}+108 \,\operatorname {expIntegral}_{1}\left (-\frac {6}{x}\right ) x -36 x \,{\mathrm e}^{\frac {6}{x}}-216 \,\operatorname {expIntegral}_{1}\left (-\frac {6}{x}\right )\right ) {\mathrm e}^{-\frac {6}{x}}}{2 x^{5}} \]

Problem 7564

ODE	\[ \boxed {\left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (1+7 x \right ) y^{\prime }+2 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x +1\right ) 2^{\frac {3}{8}} {\mathrm e}^{\frac {3 \sqrt {7}\, \arctan \left (\frac {\left (4 x +1\right ) \sqrt {7}}{7}\right )}{14}}}{\left (2 x^{2}+x +1\right )^{\frac {3}{4}}}+\frac {c_{2} \left (x +1\right ) 2^{\frac {5}{8}} {\mathrm e}^{\frac {3 \sqrt {7}\, \arctan \left (\frac {\left (4 x +1\right ) \sqrt {7}}{7}\right )}{14}} \left (\int \frac {{\mathrm e}^{-\frac {3 \sqrt {7}\, \arctan \left (\frac {\left (4 x +1\right ) \sqrt {7}}{7}\right )}{14}}}{\left (x +1\right )^{2} \left (2 x^{2}+x +1\right )^{\frac {1}{4}}}d x \right )}{2 \left (2 x^{2}+x +1\right )^{\frac {3}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (\frac {i \sqrt {7}-4 x -1}{i \sqrt {7}+4 x +1}\right )^{-\frac {3 i \sqrt {7}}{28}} \left (x +1\right )}{\left (2 x^{2}+x +1\right )^{\frac {3}{4}}}+\frac {c_{2} \left (\frac {i \sqrt {7}-4 x -1}{i \sqrt {7}+4 x +1}\right )^{-\frac {3 i \sqrt {7}}{28}} \left (x +1\right ) \left (\int \frac {\left (\frac {i \sqrt {7}+4 x +1}{i \sqrt {7}-4 x -1}\right )^{-\frac {3 i \sqrt {7}}{28}}}{\left (x +1\right )^{2} \left (2 x^{2}+x +1\right )^{\frac {1}{4}}}d x \right )}{\left (2 x^{2}+x +1\right )^{\frac {3}{4}}} \]

Problem 7565

ODE	\[ \boxed {\left (x +3\right ) y^{\prime \prime }+\left (2 x +1\right ) y^{\prime }-\left (2-x \right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{-x} x \left (6+x \right ) \left (x^{2}+9 x +27\right ) \left (x^{2}+3 x +9\right )}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{-x} \left (x^{6}+18 x^{5}+135 x^{4}+540 x^{3}+1215 x^{2}+1458 x \right ) \]

Problem 7566

ODE	\[ \boxed {y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y=0} \]
program solution	\[ y = c_{1} \left (x^{2}-1\right ) {\mathrm e}^{-x^{2}}+c_{2} \left (x^{2}-1\right ) {\mathrm e}^{-x^{2}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x^{2}-1\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{2}} \left (x^{2}-1\right )+c_{2} {\mathrm e}^{-x^{2}} \left (x^{2}-1\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x -1\right )^{2} \left (x +1\right )^{2}}d x \right ) \]

Problem 7567

ODE	\[ \boxed {\left (2+4 x \right ) y^{\prime \prime }-4 y^{\prime }-\left (4 x +6\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {2}\, {\mathrm e}^{-x}+\frac {c_{2} x \sqrt {2}\, {\mathrm e}^{x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +c_{2} x \,{\mathrm e}^{x} \]

Problem 7568

ODE	\[ \boxed {y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \left (x^{6}-15 x^{4}+45 x^{2}-15\right )+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \left (x^{6}-15 x^{4}+45 x^{2}-15\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x^{6}-15 x^{4}+45 x^{2}-15\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \left (x^{6}-15 x^{4}+45 x^{2}-15\right )+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \left (x^{6}-15 x^{4}+45 x^{2}-15\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x^{6}-15 x^{4}+45 x^{2}-15\right )^{2}}d x \right ) \]

Problem 7569

ODE	\[ \boxed {2 y^{\prime \prime }+5 x y^{\prime }+\left (2 x^{2}+4\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x^{2}}-\frac {i c_{2} {\mathrm e}^{-x^{2}} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (\frac {i \sqrt {3}\, x}{2}\right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{2}}+c_{2} {\mathrm e}^{-x^{2}} \operatorname {erf}\left (\frac {i \sqrt {3}\, x}{2}\right ) \]

Problem 7570

ODE	\[ \boxed {y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x^{2}}+c_{2} x \,{\mathrm e}^{-x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{2}}+c_{2} {\mathrm e}^{-x^{2}} x \]

Problem 7571

ODE	\[ \boxed {y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}+2\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x^{2}}+c_{2} x \,{\mathrm e}^{-x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x^{2}}+c_{2} {\mathrm e}^{-x^{2}} x \]

Problem 7572

ODE	\[ \boxed {2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y=0} \]
program solution	\[ y = \frac {2 c_{1} \sqrt {2}\, \sqrt {x^{2}+x +1}\, {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}}}{x^{2}}+\frac {c_{2} \sqrt {2}\, \sqrt {x^{2}+x +1}\, {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}} \left (\int \frac {{\mathrm e}^{\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}}}{\sqrt {x}\, \left (x^{2}+x +1\right )^{\frac {3}{2}}}d x \right )}{4 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {x^{2}+x +1}\, \left (\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}-2 x -1}\right )^{-\frac {i \sqrt {3}}{6}}}{x^{2}}+\frac {c_{2} \sqrt {x^{2}+x +1}\, \left (\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}-2 x -1}\right )^{-\frac {i \sqrt {3}}{6}} \left (\int \frac {\left (\frac {i \sqrt {3}-2 x -1}{i \sqrt {3}+2 x +1}\right )^{-\frac {i \sqrt {3}}{6}}}{\left (x^{2}+x +1\right )^{\frac {3}{2}} \sqrt {x}}d x \right )}{x^{2}} \]

Problem 7573

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{3}} {\mathrm e}^{\frac {2 x \left (x -1\right )}{3}}+c_{2} x^{\frac {1}{3}} {\mathrm e}^{\frac {2 x \left (x -1\right )}{3}} \left (\int \frac {{\mathrm e}^{-\frac {2 x \left (x -1\right )}{3}}}{x^{\frac {4}{3}}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} {\mathrm e}^{\frac {2}{3} x^{2}-\frac {2}{3} x}+c_{2} x^{\frac {1}{3}} {\mathrm e}^{\frac {2}{3} x^{2}-\frac {2}{3} x} \left (\int \frac {{\mathrm e}^{-\frac {2}{3} x^{2}+\frac {2}{3} x}}{x^{\frac {4}{3}}}d x \right ) \]

Problem 7574

ODE	\[ \boxed {12 x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{-\frac {x}{4}}}{x^{\frac {1}{4}} \left (x +1\right )^{\frac {3}{4}}}+\frac {c_{2} {\mathrm e}^{-\frac {x}{4}} \left (\int \frac {{\mathrm e}^{\frac {x}{4}}}{x^{\frac {5}{12}} \left (x +1\right )^{\frac {1}{4}}}d x \right )}{x^{\frac {1}{4}} \left (x +1\right )^{\frac {3}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-\frac {x}{4}}}{\left (x +1\right )^{\frac {3}{4}} x^{\frac {1}{4}}}+\frac {c_{2} {\mathrm e}^{-\frac {x}{4}} \left (\int \frac {{\mathrm e}^{\frac {x}{4}}}{\left (x +1\right )^{\frac {1}{4}} x^{\frac {5}{12}}}d x \right )}{\left (x +1\right )^{\frac {3}{4}} x^{\frac {1}{4}}} \]

Problem 7575

ODE	\[ \boxed {x^{2} \left (10 x^{2}+x +5\right ) y^{\prime \prime }+x \left (48 x^{2}+3 x +4\right ) y^{\prime }+\left (36 x^{2}+x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{5}} {\mathrm e}^{-\frac {2 \sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}}}{10 x^{2}+x +5}+\frac {c_{2} x^{\frac {1}{5}} {\mathrm e}^{-\frac {2 \sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}} \left (\int \frac {{\mathrm e}^{\frac {2 \sqrt {199}\, \arctan \left (\frac {\left (20 x +1\right ) \sqrt {199}}{199}\right )}{995}}}{x^{\frac {6}{5}}}d x \right )}{10 x^{2}+x +5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {1}{5}} \left (\frac {i \sqrt {199}+20 x +1}{i \sqrt {199}-20 x -1}\right )^{-\frac {i \sqrt {199}}{995}}}{10 x^{2}+x +5}+\frac {c_{2} x^{\frac {1}{5}} \left (\frac {i \sqrt {199}+20 x +1}{i \sqrt {199}-20 x -1}\right )^{-\frac {i \sqrt {199}}{995}} \left (\int \frac {\left (\frac {i \sqrt {199}-20 x -1}{i \sqrt {199}+20 x +1}\right )^{-\frac {i \sqrt {199}}{995}}}{x^{\frac {6}{5}}}d x \right )}{10 x^{2}+x +5} \]

Problem 7576

ODE	\[ \boxed {18 x^{2} \left (x +1\right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x +1\right )^{\frac {1}{6}} {\mathrm e}^{-\frac {x}{6}}}{x^{\frac {1}{6}}}+\frac {c_{2} \left (x +1\right )^{\frac {1}{6}} {\mathrm e}^{-\frac {x}{6}} \left (\int \frac {{\mathrm e}^{\frac {x}{6}}}{\sqrt {x}\, \left (x +1\right )^{\frac {7}{6}}}d x \right )}{x^{\frac {1}{6}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x}{6}} \left (\frac {x +1}{x}\right )^{\frac {1}{6}}+c_{2} {\mathrm e}^{-\frac {x}{6}} \left (\frac {x +1}{x}\right )^{\frac {1}{6}} \left (\int \frac {{\mathrm e}^{\frac {x}{6}}}{\left (x +1\right )^{\frac {7}{6}} \sqrt {x}}d x \right ) \]

Problem 7577

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (2 x +3\right ) y^{\prime }-\left (1-x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{-x}}{x}+\frac {c_{2} \left (2 \sqrt {x}-\sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {x}\right ) {\mathrm e}^{-x}\right )}{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-x}}{x}+\frac {c_{2} {\mathrm e}^{-x} \left (\int \sqrt {x}\, {\mathrm e}^{x}d x \right )}{x} \]

Problem 7578

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (5+x \right ) y^{\prime }-\left (2-3 x \right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, {\mathrm e}^{-\frac {x}{2}}+\frac {c_{2} \left (-i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \sqrt {x}}{2}\right ) x^{\frac {5}{2}} {\mathrm e}^{-\frac {x}{2}}-2 x^{2}-2 x -6\right )}{15 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, {\mathrm e}^{-\frac {x}{2}}+c_{2} \sqrt {x}\, {\mathrm e}^{-\frac {x}{2}} \left (\int \frac {{\mathrm e}^{\frac {x}{2}}}{x^{\frac {7}{2}}}d x \right ) \]

Problem 7579

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{-\frac {x}{3}}}{x^{\frac {1}{3}}}+\frac {c_{2} {\mathrm e}^{-\frac {x}{3}} \left (\int x^{\frac {1}{3}} {\mathrm e}^{\frac {x}{3}}d x \right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-\frac {x}{3}}}{x^{\frac {1}{3}}}+\frac {c_{2} {\mathrm e}^{-\frac {x}{3}} \left (\int x^{\frac {1}{3}} {\mathrm e}^{\frac {x}{3}}d x \right )}{x^{\frac {1}{3}}} \]

Problem 7580

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (1-2 x \right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, {\mathrm e}^{2 \sqrt {x}}-\frac {c_{2} \sqrt {x}\, {\mathrm e}^{-2 \sqrt {x}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \sinh \left (2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \cosh \left (2 \sqrt {x}\right ) \]

Problem 7581

ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-\left (1+3 x \right ) y=0} \]
program solution	\[ y = c_{1} \left (x^{2}+20 x +70\right ) x +c_{2} \left (x^{2}+20 x +70\right ) x \left (\int \frac {{\mathrm e}^{-\frac {x}{3}}}{x^{\frac {7}{3}} \left (x^{2}+20 x +70\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (x^{2}+20 x +70\right )+c_{2} x \left (x^{2}+20 x +70\right ) \left (\int \frac {{\mathrm e}^{-\frac {x}{3}}}{x^{\frac {7}{3}} \left (x^{2}+20 x +70\right )^{2}}d x \right ) \]

Problem 7582

ODE	\[ \boxed {2 x^{2} \left (x +3\right ) y^{\prime \prime }+x \left (1+5 x \right ) y^{\prime }+\left (x +1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{3}}}{\left (x +3\right )^{\frac {4}{3}}}+\frac {c_{2} x^{\frac {1}{3}} \left (\int \frac {\left (x +3\right )^{\frac {1}{3}}}{x^{\frac {5}{6}}}d x \right )}{\left (x +3\right )^{\frac {4}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {1}{3}}}{\left (x +3\right )^{\frac {4}{3}}}+\frac {c_{2} x^{\frac {1}{3}} \left (\int \frac {\left (x +3\right )^{\frac {1}{3}}}{x^{\frac {5}{6}}}d x \right )}{\left (x +3\right )^{\frac {4}{3}}} \]

Problem 7583

ODE	\[ \boxed {x^{2} \left (4+x \right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{4}}}{\left (4+x \right )^{\frac {9}{4}}}+\frac {c_{2} x^{\frac {1}{4}} \left (\int \frac {\left (4+x \right )^{\frac {5}{4}}}{x^{\frac {1}{4}}}d x \right )}{\left (4+x \right )^{\frac {9}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {1}{4}}}{\left (x +4\right )^{\frac {9}{4}}}+\frac {c_{2} x^{\frac {1}{4}} \left (\int \frac {\left (x +4\right )^{\frac {5}{4}}}{x^{\frac {1}{4}}}d x \right )}{\left (x +4\right )^{\frac {9}{4}}} \]

Problem 7584

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (x +1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{\sqrt {2}\, \sqrt {-x}}}{x}-\frac {c_{2} \sqrt {2}\, \sqrt {-x}\, \left ({\mathrm e}^{\sqrt {2}\, \sqrt {-x}}-{\mathrm e}^{-\sqrt {2}\, \sqrt {-x}}\right )}{2 x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (\sqrt {x}\, \sqrt {2}\right )}{x}+\frac {c_{2} \cos \left (\sqrt {x}\, \sqrt {2}\right )}{x} \]

Problem 7585

ODE	\[ \boxed {6 x^{2} y^{\prime \prime }+x \left (10-x \right ) y^{\prime }-\left (2+x \right ) y=0} \]
program solution	\[ y = \frac {c_{1}}{x}+\frac {c_{2} \left (\int x^{\frac {1}{3}} {\mathrm e}^{\frac {x}{6}}d x \right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{x}+\frac {c_{2} \left (\int x^{\frac {1}{3}} {\mathrm e}^{\frac {x}{6}}d x \right )}{x} \]

Problem 7586

ODE	\[ \boxed {x^{2} \left (4 x +3\right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (4 x +3\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (48 x^{2}+32 x +7\right ) 2^{\frac {2}{3}}}{12 x^{3}}+\frac {c_{2} \left (288 x^{2}+192 x +42\right ) 2^{\frac {1}{3}} \left (\int \frac {x^{\frac {7}{3}} \left (4 x +3\right )^{\frac {8}{3}}}{\left (48 x^{2}+32 x +7\right )^{2}}d x \right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (48 x^{2}+32 x +7\right )}{x^{3}}+\frac {c_{2} \left (48 x^{2}+32 x +7\right ) \left (\int \frac {\left (4 x +3\right )^{\frac {8}{3}} x^{\frac {7}{3}}}{\left (48 x^{2}+32 x +7\right )^{2}}d x \right )}{x^{3}} \]

Problem 7587

ODE	\[ \boxed {2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+11 x \right ) y^{\prime }-\left (1-x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} 3^{\frac {7}{12}}}{3 \sqrt {x}}+\frac {2 c_{2} 3^{\frac {5}{12}} \left (3 x +2\right )^{\frac {1}{6}}}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\sqrt {x}}+\frac {c_{2} \left (3 x +2\right )^{\frac {1}{6}}}{\sqrt {x}} \]

Problem 7588

ODE	\[ \boxed {x^{2} \left (2+x \right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (-8 x +2\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (40 x^{4}-160 x^{3}+60 x^{2}+8 x +1\right )}{40 x^{2}}+\frac {2 c_{2} \sqrt {x}\, \sqrt {2+x}\, \left (\left (-525 x^{4}+2100 x^{3}-\frac {1575}{2} x^{2}-105 x -\frac {105}{8}\right ) \ln \left (\frac {\sqrt {x \left (2+x \right )}-x}{x}\right )+\left (525 x^{4}-2100 x^{3}+\frac {1575}{2} x^{2}+105 x +\frac {105}{8}\right ) \ln \left (\frac {x +\sqrt {x \left (2+x \right )}}{x}\right )+\sqrt {x \left (2+x \right )}\, \left (x^{5}+41 x^{4}-\frac {6987}{4} x^{3}+\frac {13367}{4} x^{2}-\frac {805}{8} x -\frac {105}{8}\right )\right )}{\sqrt {x \left (2+x \right )}\, \left (x +\sqrt {x \left (2+x \right )}\right )^{2} \left (-\sqrt {x \left (2+x \right )}+x \right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (40 x^{4}-160 x^{3}+60 x^{2}+8 x +1\right )}{x^{2}}+\frac {c_{2} \left (40 x^{4}-160 x^{3}+60 x^{2}+8 x +1\right ) \left (\int \frac {x^{\frac {3}{2}} \left (x +2\right )^{\frac {15}{2}}}{\left (40 x^{4}-160 x^{3}+60 x^{2}+8 x +1\right )^{2}}d x \right )}{x^{2}} \]

Problem 7589

ODE	\[ \boxed {8 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-13 x^{2}+1\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {1}{4}}}{\left (x +1\right )^{\frac {3}{4}} \left (x -1\right )^{\frac {3}{4}}}+\frac {c_{2} \left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {1}{4}} \left (\int \frac {1}{x^{\frac {3}{4}} \sqrt {x^{2}-1}}d x \right )}{\left (x +1\right )^{\frac {3}{4}} \left (x -1\right )^{\frac {3}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {\frac {1}{\left (x -1\right ) \left (x +1\right )}}\, x^{\frac {1}{4}}+c_{2} \sqrt {\frac {1}{\left (x -1\right ) \left (x +1\right )}}\, x^{\frac {1}{4}} \left (\int \frac {\sqrt {\frac {1}{\left (x -1\right ) \left (x +1\right )}}}{x^{\frac {3}{4}}}d x \right ) \]

Problem 7590

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-2 x \left (-x^{2}+2\right ) y^{\prime }+4 y=0} \]
program solution	\[ y = \frac {c_{1} x^{4}}{\left (x^{2}+1\right )^{2}}+\frac {c_{2} \left (-3 x^{3}-x \right )}{3 \left (x^{2}+1\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (3 x^{2}+1\right )}{\left (x^{2}+1\right )^{2}}+\frac {c_{2} x^{4}}{\left (x^{2}+1\right )^{2}} \]

Problem 7591

ODE	\[ \boxed {x \left (x^{2}+3\right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }-8 y x=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (x^{2}+3\right )^{\frac {11}{6}}+c_{2} \left (-\frac {8}{55} x^{4}-\frac {4}{5} x^{2}-1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{4}+\frac {11}{2} x^{2}+\frac {55}{8}\right )+c_{2} \left (x^{2}+3\right )^{\frac {11}{6}} x^{\frac {1}{3}} \]

Problem 7592

ODE	\[ \boxed {4 x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+x \left (-19 x^{2}+7\right ) y^{\prime }-\left (14 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}-1\right )^{\frac {1}{4}}}{x \left (x +1\right )^{\frac {3}{4}} \left (x -1\right )^{\frac {3}{4}}}+\frac {c_{2} \left (x^{2}-1\right )^{\frac {1}{4}} \left (\int \frac {x^{\frac {1}{4}}}{\sqrt {x^{2}-1}}d x \right )}{x \left (x +1\right )^{\frac {3}{4}} \left (x -1\right )^{\frac {3}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {\frac {1}{\left (x -1\right ) \left (x +1\right )}}}{x}+\frac {c_{2} \sqrt {\frac {1}{\left (x -1\right ) \left (x +1\right )}}\, \left (\int \sqrt {\frac {1}{\left (x -1\right ) \left (x +1\right )}}\, x^{\frac {1}{4}}d x \right )}{x} \]

Problem 7593

ODE	\[ \boxed {3 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+x \left (-11 x^{2}+1\right ) y^{\prime }+\left (-5 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \sqrt {x}}{\left (x^{2}-2\right )^{\frac {3}{4}}}+\frac {c_{2} \sqrt {x}\, \left (\int \frac {1}{\left (x^{2}-2\right )^{\frac {1}{4}} x^{\frac {7}{6}}}d x \right )}{\left (x^{2}-2\right )^{\frac {3}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {x}}{\left (x^{2}-2\right )^{\frac {3}{4}}}+\frac {c_{2} \sqrt {x}\, \left (\int \frac {1}{\left (x^{2}-2\right )^{\frac {1}{4}} x^{\frac {7}{6}}}d x \right )}{\left (x^{2}-2\right )^{\frac {3}{4}}} \]

Problem 7594

ODE	\[ \boxed {2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {7}{2}}}{\left (x^{2}+2\right )^{\frac {9}{4}}}+\frac {c_{2} x^{\frac {7}{2}} \left (\int \frac {\left (x^{2}+2\right )^{\frac {5}{4}}}{x^{4}}d x \right )}{\left (x^{2}+2\right )^{\frac {9}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {7}{2}}}{\left (x^{2}+2\right )^{\frac {9}{4}}}+\frac {c_{2} x^{\frac {7}{2}} \left (\int \frac {\left (x^{2}+2\right )^{\frac {5}{4}}}{x^{4}}d x \right )}{\left (x^{2}+2\right )^{\frac {9}{4}}} \]

Problem 7595

ODE	\[ \boxed {2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (7 x^{2}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1}}{\left (x^{2}+2\right )^{\frac {1}{4}} \sqrt {x}}+\frac {c_{2} \left (\int \frac {1}{\left (x^{2}+2\right )^{\frac {3}{4}}}d x \right )}{\left (x^{2}+2\right )^{\frac {1}{4}} \sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\left (x^{2}+2\right )^{\frac {1}{4}} \sqrt {x}}+\frac {c_{2} \left (\int \frac {1}{\left (x^{2}+2\right )^{\frac {3}{4}}}d x \right )}{\left (x^{2}+2\right )^{\frac {1}{4}} \sqrt {x}} \]

Problem 7596

ODE	\[ \boxed {2 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+5 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-40 x^{2}+2\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {3}{4}} 2^{\frac {3}{4}}}{2 \left (2 x^{2}+1\right )^{\frac {5}{4}} \left (2 x^{3}+x \right )^{\frac {1}{4}}}+\frac {c_{2} x^{\frac {3}{4}} 2^{\frac {1}{4}} \left (\int \frac {\sqrt {2 x^{2}+1}}{x^{\frac {7}{2}}}d x \right )}{\left (2 x^{2}+1\right )^{\frac {5}{4}} \left (2 x^{3}+x \right )^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {x}}{\left (2 x^{2}+1\right )^{\frac {3}{2}}}+\frac {c_{2} \sqrt {x}\, \left (\int \frac {\sqrt {2 x^{2}+1}}{x^{\frac {7}{2}}}d x \right )}{\left (2 x^{2}+1\right )^{\frac {3}{2}}} \]

Problem 7597

ODE	\[ \boxed {x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 y x=0} \]
program solution	\[ y = \frac {c_{1}}{\sqrt {x^{2}+1}\, x^{3}}+\frac {c_{2} \left (x^{3}-\operatorname {arcsinh}\left (x \right ) \sqrt {x^{2}+1}+x \right )}{2 \left (x^{2}+1\right ) x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\sqrt {x^{2}+1}\, x^{3}}+\frac {c_{2} \left (\frac {x \sqrt {x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (x \right )}{2}\right )}{\sqrt {x^{2}+1}\, x^{3}} \]

Problem 7598

ODE	\[ \boxed {2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y=0} \]
program solution	\[ y = \frac {c_{1}}{\left (x^{2}+1\right )^{\frac {1}{4}} x^{\frac {3}{2}}}+\frac {c_{2} \left (\int \frac {x^{\frac {3}{2}}}{\left (x^{2}+1\right )^{\frac {3}{4}}}d x \right )}{\left (x^{2}+1\right )^{\frac {1}{4}} x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\left (x^{2}+1\right )^{\frac {1}{4}} x^{\frac {3}{2}}}+\frac {c_{2} \left (\int \frac {x^{\frac {3}{2}}}{\left (x^{2}+1\right )^{\frac {3}{4}}}d x \right )}{\left (x^{2}+1\right )^{\frac {1}{4}} x^{\frac {3}{2}}} \]

Problem 7599

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{-\frac {x^{2}}{6}}}{x^{\frac {1}{3}}}+\frac {c_{2} {\mathrm e}^{-\frac {x^{2}}{6}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{6}}}{x^{\frac {1}{3}}}d x \right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-\frac {x^{2}}{6}}}{x^{\frac {1}{3}}}+\frac {c_{2} {\mathrm e}^{-\frac {x^{2}}{6}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{6}}}{x^{\frac {1}{3}}}d x \right )}{x^{\frac {1}{3}}} \]

Problem 7600

ODE	\[ \boxed {6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x^{\frac {5}{6}}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x^{\frac {5}{6}}}d x \right ) \]