2.17.81 Problems 8001 to 8100

Problem 8001

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 8002

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 8003

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 8004

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 8005

ODE	\[ \boxed {x^{2} \left (4+x \right ) y^{\prime \prime }-x \left (-3 x +1\right ) y^{\prime }+y=0} \]
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 8006

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 8007

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 8008

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 8009

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 8010

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 8011

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 8012

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	\[ y \left (x \right ) = \frac {c_{1} x \left (3 x^{2}+1\right )}{\left (x^{2}+1\right )^{2}}+\frac {x^{4} c_{2}}{\left (x^{2}+1\right )^{2}} \]

Problem 8013

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 8014

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 8015

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 8016

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 8017

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 8018

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	\[ y = \frac {c_{1} 2^{\frac {3}{4}} x^{\frac {3}{4}}}{2 \left (2 x^{2}+1\right )^{\frac {5}{4}} \left (2 x^{3}+x \right )^{\frac {1}{4}}}+\frac {c_{2} x^{\frac {3}{4}} 2^{\frac {1}{4}} \left (\int \frac {\sqrt {2 x^{2}+1}}{x^{\frac {7}{2}}}d x \right )}{\left (2 x^{2}+1\right )^{\frac {5}{4}} \left (2 x^{3}+x \right )^{\frac {1}{4}}} \] 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 8019

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 8020

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 8021

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 8022

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	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} x^{\frac {1}{3}} {\mathrm e}^{-\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x^{\frac {5}{6}}}d x \right ) \]

Problem 8023

ODE	\[ \boxed {9 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+3\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{\frac {1}{3}} \left (x^{2}+1\right )^{\frac {2}{3}}}+\frac {c_{2} \left (\int \frac {1}{\left (x^{2}+1\right )^{\frac {1}{3}} x^{\frac {1}{3}}}d x \right )}{x^{\frac {1}{3}} \left (x^{2}+1\right )^{\frac {2}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\left (x^{2}+1\right )^{\frac {2}{3}} x^{\frac {1}{3}}}+\frac {c_{2} \left (\int \frac {1}{\left (x^{3}+x \right )^{\frac {1}{3}}}d x \right )}{\left (x^{2}+1\right )^{\frac {2}{3}} x^{\frac {1}{3}}} \]

Problem 8024

ODE	\[ \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \sqrt {x}}{\left (x^{2}+1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (\operatorname {arcsinh}\left (x \right ) x -\sqrt {x^{2}+1}\right )}{\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {x}}{\left (x^{2}+1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (\operatorname {arcsinh}\left (x \right ) x -\sqrt {x^{2}+1}\right )}{\sqrt {x}\, \left (x^{2}+1\right )^{\frac {3}{2}}} \]

Problem 8025

ODE	\[ \boxed {8 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+2 x \left (34 x^{2}+5\right ) y^{\prime }-\left (-30 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{4}} 2^{\frac {1}{4}}}{\sqrt {2 x^{2}+1}}+\frac {c_{2} x^{\frac {1}{4}} 2^{\frac {3}{4}} \left (\int \frac {1}{\sqrt {2 x^{2}+1}\, x^{\frac {7}{4}}}d x \right )}{2 \sqrt {2 x^{2}+1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {1}{4}}}{\sqrt {2 x^{2}+1}}+\frac {c_{2} x^{\frac {1}{4}} \left (\int \frac {1}{\sqrt {2 x^{2}+1}\, x^{\frac {7}{4}}}d x \right )}{\sqrt {2 x^{2}+1}} \]

Problem 8026

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

Problem 8027

ODE	\[ \boxed {6 x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (50 x^{2}+1\right ) y^{\prime }+\left (30 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{3}}}{2 x^{2}+1}+\frac {6 c_{2} \sqrt {x}}{2 x^{2}+1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {x}}{2 x^{2}+1}+\frac {c_{2} x^{\frac {1}{3}}}{2 x^{2}+1} \]

Problem 8028

ODE	\[ \boxed {28 x^{2} \left (-3 x +1\right ) y^{\prime \prime }-7 x \left (5+9 x \right ) y^{\prime }+7 \left (2+9 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{4}}}{3 x -1}+\frac {4 c_{2} x^{2}}{21 x -7} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2}}{3 x -1}+\frac {c_{2} x^{\frac {1}{4}}}{3 x -1} \]

Problem 8029

ODE	\[ \boxed {8 x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+2 x \left (-21 x^{2}+10\right ) y^{\prime }-\left (35 x^{2}+2\right ) y=0} \]
program solution	\[ y = \frac {c_{1}}{\sqrt {x}\, \left (x^{2}-2\right )}+\frac {4 c_{2} x^{\frac {1}{4}}}{3 x^{2}-6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\left (x^{2}-2\right ) \sqrt {x}}+\frac {c_{2} x^{\frac {1}{4}}}{x^{2}-2} \]

Problem 8030

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

Problem 8031

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

Problem 8032

ODE	\[ \boxed {4 x^{2} \left (x^{2}+2 x +3\right ) y^{\prime \prime }-x \left (-15 x^{2}-14 x +3\right ) y^{\prime }+\left (7 x^{2}+3\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{4}}}{x^{2}+2 x +3}+\frac {4 c_{2} x}{3 x^{2}+6 x +9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x}{x^{2}+2 x +3}+\frac {c_{2} x^{\frac {1}{4}}}{x^{2}+2 x +3} \]

Problem 8033

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

Problem 8034

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

Problem 8035

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

Problem 8036

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

Problem 8037

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

Problem 8038

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

Problem 8039

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{\frac {x \left (x -3\right )}{3}}}{x^{\frac {1}{3}}}+\frac {c_{2} {\mathrm e}^{\frac {x \left (x -3\right )}{3}} \left (\int \frac {{\mathrm e}^{-\frac {x \left (x -3\right )}{3}}}{x}d x \right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\frac {1}{3} x^{2}-x}}{x^{\frac {1}{3}}}+\frac {c_{2} {\mathrm e}^{\frac {1}{3} x^{2}-x} \left (\int \frac {{\mathrm e}^{-\frac {1}{3} x^{2}+x}}{x}d x \right )}{x^{\frac {1}{3}}} \]

Problem 8040

ODE	\[ \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y=0} \]
program solution	\[ y = \frac {c_{1} 2^{\frac {5}{8}} {\mathrm e}^{-\frac {3 x}{2}}}{2 x^{2} \left (2 x +1\right )^{\frac {1}{4}}}+\frac {c_{2} 2^{\frac {3}{8}} {\mathrm e}^{-\frac {3 x}{2}} \left (\int \frac {{\mathrm e}^{\frac {3 x}{2}}}{x \left (2 x +1\right )^{\frac {3}{4}}}d x \right )}{x^{2} \left (2 x +1\right )^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-\frac {3 x}{2}}}{\left (2 x +1\right )^{\frac {1}{4}} x^{2}}+\frac {c_{2} {\mathrm e}^{-\frac {3 x}{2}} \left (\int \frac {{\mathrm e}^{\frac {3 x}{2}}}{\left (2 x +1\right )^{\frac {3}{4}} x}d x \right )}{\left (2 x +1\right )^{\frac {1}{4}} x^{2}} \]

Problem 8041

ODE	\[ \boxed {16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{-\frac {x \left (x +1\right )}{4}}}{x^{\frac {1}{4}}}+\frac {c_{2} {\mathrm e}^{-\frac {x \left (x +1\right )}{4}} \left (\int \frac {{\mathrm e}^{\frac {x \left (x +1\right )}{4}}}{x}d x \right )}{x^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-\frac {1}{4} x^{2}-\frac {1}{4} x}}{x^{\frac {1}{4}}}+\frac {c_{2} {\mathrm e}^{-\frac {1}{4} x^{2}-\frac {1}{4} x} \left (\int \frac {{\mathrm e}^{\frac {1}{4} x^{2}+\frac {1}{4} x}}{x}d x \right )}{x^{\frac {1}{4}}} \]

Problem 8042

ODE	\[ \boxed {9 x^{2} \left (x +1\right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{\frac {x}{3}}}{x^{\frac {1}{3}} \left (x +1\right )^{\frac {4}{3}}}+\frac {c_{2} {\mathrm e}^{\frac {x}{3}} \left (\int \frac {\left (x +1\right )^{\frac {1}{3}} {\mathrm e}^{-\frac {x}{3}}}{x}d x \right )}{x^{\frac {1}{3}} \left (x +1\right )^{\frac {4}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\frac {x}{3}}}{\left (x +1\right )^{\frac {4}{3}} x^{\frac {1}{3}}}+\frac {c_{2} {\mathrm e}^{\frac {x}{3}} \left (\int \frac {\left (x +1\right )^{\frac {1}{3}} {\mathrm e}^{-\frac {x}{3}}}{x}d x \right )}{\left (x +1\right )^{\frac {4}{3}} x^{\frac {1}{3}}} \]

Problem 8043

ODE	\[ \boxed {36 x^{2} \left (1-2 x \right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{6}} 2^{\frac {1}{6}}}{\left (2 x -1\right )^{\frac {4}{3}}}-\frac {c_{2} x^{\frac {1}{6}} 2^{\frac {5}{6}} \left (\sqrt {3}\, \arctan \left (\frac {\left (2 \left (2 x -1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )-3 \left (2 x -1\right )^{\frac {1}{3}}+\ln \left (\left (2 x -1\right )^{\frac {1}{3}}+1\right )-\frac {\ln \left (\left (2 x -1\right )^{\frac {2}{3}}-\left (2 x -1\right )^{\frac {1}{3}}+1\right )}{2}\right )}{2 \left (2 x -1\right )^{\frac {4}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {1}{6}}}{\left (-1+2 x \right )^{\frac {4}{3}}}+\frac {c_{2} x^{\frac {1}{6}} \left (\int \frac {\left (-1+2 x \right )^{\frac {1}{3}}}{x}d x \right )}{\left (-1+2 x \right )^{\frac {4}{3}}} \]

Problem 8044

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

Problem 8045

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

Problem 8046

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

Problem 8047

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

Problem 8048

ODE	\[ \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (5+9 x \right ) y^{\prime }+\left (4+3 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} 2^{\frac {3}{4}} \left (2 x +1\right )^{\frac {3}{2}}}{4 x^{2}}+\frac {4 c_{2} 2^{\frac {1}{4}} \left (\sqrt {2 x +1}\, \left (x +\frac {1}{2}\right ) \ln \left (\sqrt {2 x +1}-1\right )-\sqrt {2 x +1}\, \left (x +\frac {1}{2}\right ) \ln \left (\sqrt {2 x +1}+1\right )+2 x +\frac {4}{3}\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (2 x +1\right )^{\frac {3}{2}}}{x^{2}}+\frac {c_{2} \left (-12 \ln \left (\sqrt {2 x +1}+1\right ) x^{2}-12 \ln \left (\sqrt {2 x +1}+1\right ) x +12 \ln \left (\sqrt {2 x +1}-1\right ) x^{2}+12 \ln \left (\sqrt {2 x +1}-1\right ) x -3 \ln \left (\sqrt {2 x +1}+1\right )+3 \ln \left (\sqrt {2 x +1}-1\right )+12 \sqrt {2 x +1}\, x +8 \sqrt {2 x +1}\right )}{3 x^{2} \sqrt {2 x +1}} \]

Problem 8049

ODE	\[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (4 x +5\right ) y^{\prime }+\left (9+4 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (8 x +1\right ) x^{3} \sqrt {2}}{2 \left (2 x -1\right )^{6}}+\frac {4 c_{2} \sqrt {2}\, x^{3} \left (\left (-6 x -\frac {3}{4}\right ) \ln \left (x \right )+x^{4}-4 x^{3}+9 x^{2}+\frac {609 x}{512}-\frac {9375}{4096}\right )}{3 \left (2 x -1\right )^{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (8 x +1\right )}{\left (-1+2 x \right )^{6}}+\frac {c_{2} x^{3} \left (\frac {4 x^{4}}{3}-\frac {16 x^{3}}{3}-8 x \ln \left (x \right )+12 x^{2}-\ln \left (x \right )+\frac {203 x}{128}-\frac {3125}{1024}\right )}{\left (-1+2 x \right )^{6}} \]

Problem 8050

ODE	\[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{4}+16 x^{3}+36 x^{2}+16 x +1\right )}{x^{3}}+\frac {c_{2} \left (25+3 \ln \left (x \right ) \left (x^{4}+16 x^{3}+36 x^{2}+16 x +1\right )+120 x^{3}+450 x^{2}+280 x \right )}{3 x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{4}+16 x^{3}+36 x^{2}+16 x +1\right )}{x^{3}}+\frac {c_{2} \left (x^{4} \ln \left (x \right )+16 x^{3} \ln \left (x \right )+36 x^{2} \ln \left (x \right )+40 x^{3}+16 x \ln \left (x \right )+150 x^{2}+\ln \left (x \right )+\frac {280 x}{3}+\frac {25}{3}\right )}{x^{3}} \]

Problem 8051

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

Problem 8052

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

Problem 8053

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

Problem 8054

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

Problem 8055

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

Problem 8056

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

Problem 8057

ODE	\[ \boxed {4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}+4\right )^{\frac {5}{8}}}{x^{\frac {1}{4}}}+\frac {c_{2} \left (x^{2}+4\right )^{\frac {5}{8}} \left (\int \frac {1}{\left (x^{2}+4\right )^{\frac {13}{8}} x}d x \right )}{x^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+4\right )^{\frac {5}{8}}}{x^{\frac {1}{4}}}+\frac {c_{2} \left (x^{2}+4\right )^{\frac {5}{8}} \left (\int \frac {1}{\left (x^{2}+4\right )^{\frac {13}{8}} x}d x \right )}{x^{\frac {1}{4}}} \]

Problem 8058

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

Problem 8059

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }-3 x \left (-2 x^{2}+7\right ) y^{\prime }+\left (2 x^{2}+25\right ) y=0} \]
program solution	\[ y = c_{1} x^{\frac {5}{3}} {\mathrm e}^{-\frac {x^{2}}{3}}-\frac {c_{2} x^{\frac {5}{3}} {\mathrm e}^{-\frac {x^{2}}{3}} \operatorname {expIntegral}_{1}\left (-\frac {x^{2}}{3}\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {5}{3}} {\mathrm e}^{-\frac {x^{2}}{3}}+c_{2} x^{\frac {5}{3}} {\mathrm e}^{-\frac {x^{2}}{3}} \operatorname {expIntegral}_{1}\left (-\frac {x^{2}}{3}\right ) \]

Problem 8060

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

Problem 8061

ODE	\[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }+3 x y^{\prime }+\left (4 x +1\right ) y=0} \]
program solution	\[ y = \frac {2 c_{1} \sqrt {2}}{x}-\frac {c_{2} \sqrt {2}\, \left (-16 x^{3}+36 x^{2}+6 \ln \left (x \right )-36 x -3\right )}{24 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{x}-\frac {c_{2} \left (-8 x^{3}+18 x^{2}+3 \ln \left (x \right )-18 x \right )}{3 x} \]

Problem 8062

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

Problem 8063

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

Problem 8064

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\right ) y=0} \]
program solution	\[ y = c_{1} \left (x^{5}-4 x^{3}+x \right )+c_{2} \left (\ln \left (x \right ) \left (x^{4}-4 x^{2}+1\right )-6 x^{2}+3\right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (x^{4}-4 x^{2}+1\right )+c_{2} \left (x^{4} \ln \left (x \right )-4 x^{2} \ln \left (x \right )-6 x^{2}+\ln \left (x \right )+3\right ) x \]

Problem 8065

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

Problem 8066

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

Problem 8067

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{4}-16 x^{2}+32\right )}{\sqrt {x}}+\frac {c_{2} \left (x^{4}-16 x^{2}+32\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{x \left (x^{4}-16 x^{2}+32\right )^{2}}d x \right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{4}-16 x^{2}+32\right )}{\sqrt {x}}+\frac {c_{2} \left (x^{4}-16 x^{2}+32\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{4}}}{x \left (x^{4}-16 x^{2}+32\right )^{2}}d x \right )}{\sqrt {x}} \]

Problem 8068

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

Problem 8069

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

Problem 8070

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

Problem 8071

ODE	\[ \boxed {16 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x \left (9 x^{2}+1\right ) y^{\prime }+\left (49 x^{2}+1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{4}}}{x^{2}+1}+\frac {c_{2} x^{\frac {1}{4}} \left (2 \ln \left (x \right )-1\right )}{2 x^{2}+2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {1}{4}}}{x^{2}+1}+\frac {c_{2} x^{\frac {1}{4}} \ln \left (x \right )}{x^{2}+1} \]

Problem 8072

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

Problem 8073

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

Problem 8074

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

Problem 8075

ODE	\[ \boxed {9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {1}{3}}}{x^{2}+x +1}+\frac {c_{2} x^{\frac {1}{3}} \left (24 \ln \left (x \right )+13\right )}{24 x^{2}+24 x +24} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {1}{3}}}{x^{2}+x +1}+\frac {c_{2} x^{\frac {1}{3}} \ln \left (x \right )}{x^{2}+x +1} \]

Problem 8076

ODE	\[ \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {5}{2}}}{\left (2+x \right )^{\frac {7}{2}}}+\frac {c_{2} \left (-15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2+x}\, \sqrt {2}}{2}\right ) x^{3}-66 x^{2} \sqrt {2+x}-104 \sqrt {2+x}\, x -64 \sqrt {2+x}\right )}{48 \sqrt {x}\, \left (2+x \right )^{\frac {7}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\frac {5}{2}}}{\left (x +2\right )^{\frac {7}{2}}}-\frac {c_{2} \sqrt {2}\, \left (33 \sqrt {2}\, \sqrt {x +2}\, x^{2}+15 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x +2}}{2}\right ) x^{3}+52 \sqrt {2}\, \sqrt {x +2}\, x +32 \sqrt {2}\, \sqrt {x +2}\right )}{48 \sqrt {x}\, \left (x +2\right )^{\frac {7}{2}}} \]

Problem 8077

ODE	\[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (4+3 x \right ) 2^{\frac {1}{4}} \left (2 x -1\right )^{\frac {9}{2}}}{24 x^{6}}-\frac {4 c_{2} \left (231 x^{3}-198 x^{2}+66 x -8\right ) 2^{\frac {3}{4}}}{1155 x^{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (231 x^{3}-198 x^{2}+66 x -8\right )}{x^{6}}+\frac {c_{2} \left (3 x +4\right ) \left (-1+2 x \right )^{\frac {9}{2}}}{x^{6}} \]

Problem 8078

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (10 x^{2}+3\right ) y^{\prime }-\left (-14 x^{2}+15\right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{3}}{\left (x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 c_{2} \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right ) x^{8}-\left (x^{4}-\frac {8}{3} x^{2}-\frac {8}{3}\right ) \sqrt {x^{2}+1}\, \left (x^{2}+2\right )\right )}{128 \left (x^{2}+1\right )^{\frac {5}{2}} x^{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3}}{\left (x^{2}+1\right )^{\frac {5}{2}}}-\frac {c_{2} \left (-3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right ) x^{8}+3 \sqrt {x^{2}+1}\, x^{6}-2 x^{4} \sqrt {x^{2}+1}-24 \sqrt {x^{2}+1}\, x^{2}-16 \sqrt {x^{2}+1}\right )}{128 x^{5} \left (x^{2}+1\right )^{\frac {5}{2}}} \]

Problem 8079

ODE	\[ \boxed {x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 y x^{2}=0} \]
program solution	\[ y = \frac {c_{1} \left (2 x^{2}-1\right )^{\frac {5}{4}}}{x^{6}}+\frac {c_{2} \left (5 x^{4}-20 x^{2}+8\right )}{15 x^{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (5 x^{4}-20 x^{2}+8\right )}{x^{6}}+\frac {c_{2} \left (2 x^{2}-1\right )^{\frac {5}{4}}}{x^{6}} \]

Problem 8080

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

Problem 8081

ODE	\[ \boxed {2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \sqrt {x}\, 3^{\frac {1}{4}}}{\left (3 x +2\right )^{\frac {3}{2}}}-\frac {c_{2} 3^{\frac {3}{4}} \left (\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {3 x +2}\, \sqrt {2}}{2}\right ) x +\frac {2 \sqrt {3 x +2}}{3}\right )}{2 \sqrt {x}\, \left (3 x +2\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {x}}{\left (3 x +2\right )^{\frac {3}{2}}}+\frac {c_{2} \sqrt {2}\, \left (\sqrt {2}\, \sqrt {3 x +2}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {3 x +2}}{2}\right ) x \right )}{2 \sqrt {x}\, \left (3 x +2\right )^{\frac {3}{2}}} \]

Problem 8082

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

Problem 8083

ODE	\[ \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 x \left (3+8 x \right ) y^{\prime }-\left (5-49 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \sqrt {x}}{\left (x +1\right )^{4}}+\frac {c_{2} \left (6 \ln \left (x \right ) x^{3}-18 x^{2}-9 x -2\right )}{6 x^{\frac {5}{2}} \left (x +1\right )^{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sqrt {x}}{\left (x +1\right )^{4}}+\frac {c_{2} \left (6 x^{3} \ln \left (x \right )-18 x^{2}-9 x -2\right )}{6 \left (x +1\right )^{4} x^{\frac {5}{2}}} \]

Problem 8084

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 y x=0} \]
program solution	\[ y = c_{1} \left (x^{5}-\frac {5}{2} x^{4}\right )+c_{2} \left (12 \ln \left (x \right ) x^{5}+x^{6}-30 \ln \left (x \right ) x^{4}-\frac {5 x^{5}}{2}-\frac {299 x^{4}}{4}+20 x^{3}+5 x^{2}+x +\frac {1}{10}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{5}-\frac {5}{2} x^{4}\right )+c_{2} \left (3 x^{5} \ln \left (x \right )+\frac {x^{6}}{4}-\frac {15 x^{4} \ln \left (x \right )}{2}-\frac {5 x^{5}}{8}-\frac {299 x^{4}}{16}+5 x^{3}+\frac {5 x^{2}}{4}+\frac {x}{4}+\frac {1}{40}\right ) \]

Problem 8085

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-3 \left (x +3\right ) y=0} \]
program solution	\[ y = c_{1} x^{3}+\frac {c_{2} \left (\left (x^{5}-x^{4}+2 x^{3}-6 x^{2}+24 x -120\right ) {\mathrm e}^{-x}-\operatorname {expIntegral}_{1}\left (x \right ) x^{6}\right )}{720 x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3}-\frac {c_{2} \left (-\operatorname {expIntegral}_{1}\left (x \right ) x^{6}+{\mathrm e}^{-x} x^{5}-{\mathrm e}^{-x} x^{4}+2 \,{\mathrm e}^{-x} x^{3}-6 x^{2} {\mathrm e}^{-x}+24 \,{\mathrm e}^{-x} x -120 \,{\mathrm e}^{-x}\right )}{720 x^{3}} \]

Problem 8086

ODE	\[ \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y=0} \]
program solution	\[ y = \frac {2 c_{1} \left (143 x^{2}+104 x +20\right ) 2^{\frac {1}{4}}}{143 x^{7}}+\frac {c_{2} \left (2 x +1\right )^{\frac {7}{2}} \left (35 x^{3}-45 x^{2}+36 x -20\right ) 2^{\frac {3}{4}}}{1260 x^{7}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (143 x^{2}+104 x +20\right )}{x^{7}}+\frac {c_{2} \left (35 x^{3}-45 x^{2}+36 x -20\right ) \left (2 x +1\right )^{\frac {7}{2}}}{x^{7}} \]

Problem 8087

ODE	\[ \boxed {4 x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (-x +4\right ) y^{\prime }-\left (7+5 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} 2^{\frac {7}{8}}}{4 \sqrt {x}}+\frac {4 c_{2} 2^{\frac {1}{8}} \left (x^{3}-2 x^{2}-8 x -\frac {16}{5}\right )}{7 \sqrt {x}\, \left (2 x +1\right )^{\frac {5}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\sqrt {x}}+\frac {c_{2} \left (5 x^{3}-10 x^{2}-40 x -16\right )}{\left (2 x +1\right )^{\frac {5}{4}} \sqrt {x}} \]

Problem 8088

ODE	\[ \boxed {3 x^{2} \left (x +3\right ) y^{\prime \prime }-x \left (15+x \right ) y^{\prime }-20 y=0} \]
program solution	\[ y = \frac {c_{1} \left (7 x +27\right )}{7 x^{\frac {2}{3}} \left (x +3\right )^{\frac {1}{3}}}+\frac {3 c_{2} \left (x^{2}-36 x -243\right )}{4 x^{\frac {2}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{2}-36 x -243\right )}{x^{\frac {2}{3}}}+\frac {c_{2} \left (7 x +27\right )}{x^{\frac {2}{3}} \left (x +3\right )^{\frac {1}{3}}} \]

Problem 8089

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x -\frac {5}{8}\right ) \left (x +1\right )^{12}}{x^{3}}+\frac {c_{2} \left (-715 x^{4}-572 x^{3}-234 x^{2}-52 x -5\right )}{6435 x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (715 x^{4}+572 x^{3}+234 x^{2}+52 x +5\right )}{x^{3}}+\frac {c_{2} \left (x^{13}+\frac {91}{8} x^{12}+\frac {117}{2} x^{11}+\frac {715}{4} x^{10}+\frac {715}{2} x^{9}+\frac {3861}{8} x^{8}+429 x^{7}+\frac {429}{2} x^{6}\right )}{x^{3}} \]

Problem 8090

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+3 x^{2} y^{\prime }-\left (6-x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (4+x \right )}{x^{2}}+\frac {c_{2} \left (6 \left (x +1\right )^{2} \left (4+x \right ) \ln \left (x +1\right )+60 x^{2}+129 x +68\right )}{6 x^{2} \left (x +1\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{3}+6 x^{2}+9 x +4\right )}{x^{2} \left (x +1\right )^{2}}+\frac {c_{2} \left (\ln \left (x +1\right ) x^{3}+6 \ln \left (x +1\right ) x^{2}+9 \ln \left (x +1\right ) x +10 x^{2}+4 \ln \left (x +1\right )+\frac {43 x}{2}+\frac {34}{3}\right )}{x^{2} \left (x +1\right )^{2}} \]

Problem 8091

ODE	\[ \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (3+14 x \right ) y^{\prime }+\left (6+100 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} x \left (2 x +1\right )^{9}}{32}+c_{2} \left (-\frac {16}{5} x^{5}-\frac {16}{15} x^{4}-\frac {8}{35} x^{3}-\frac {1}{35} x^{2}-\frac {1}{630} x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (2016 x^{4}+672 x^{3}+144 x^{2}+18 x +1\right )+c_{2} x \left (x^{9}+\frac {9}{2} x^{8}+9 x^{7}+\frac {21}{2} x^{6}+\frac {63}{8} x^{5}\right ) \]

Problem 8092

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }-x \left (6+11 x \right ) y^{\prime }+\left (6+32 x \right ) y=0} \]
program solution	\[ y = c_{1} \left (x -\frac {4}{3}\right ) x \left (x +1\right )^{6}+c_{2} \left (-\frac {1}{4} x^{4}-\frac {3}{10} x^{3}-\frac {3}{20} x^{2}-\frac {1}{35} x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (35 x^{3}+42 x^{2}+21 x +4\right )+c_{2} x \left (x^{7}+\frac {14}{3} x^{6}+7 x^{5}\right ) \]

Problem 8093

ODE	\[ \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 x \left (4 x +1\right ) y^{\prime }-\left (49+27 x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} x^{\frac {7}{2}}}{\left (x +1\right )^{2}}+\frac {c_{2} \left (-7 x -6\right )}{42 x^{\frac {7}{2}} \left (x +1\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (7 x +6\right )}{\left (x +1\right )^{2} x^{\frac {7}{2}}}+\frac {c_{2} x^{\frac {7}{2}}}{\left (x +1\right )^{2}} \]

Problem 8094

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

Problem 8095

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (-x^{2}+7\right ) y^{\prime }+12 y=0} \]
program solution	\[ y = c_{1} x^{6} {\mathrm e}^{-\frac {x^{2}}{2}}-\frac {c_{2} x^{2} \left (\operatorname {expIntegral}_{1}\left (-\frac {x^{2}}{2}\right ) x^{4} {\mathrm e}^{-\frac {x^{2}}{2}}+2 x^{2}+4\right )}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{6} {\mathrm e}^{-\frac {x^{2}}{2}}+\frac {c_{2} x^{2} {\mathrm e}^{-\frac {x^{2}}{2}} \left (\operatorname {expIntegral}_{1}\left (-\frac {x^{2}}{2}\right ) x^{4}+2 \,{\mathrm e}^{\frac {x^{2}}{2}} x^{2}+4 \,{\mathrm e}^{\frac {x^{2}}{2}}\right )}{16} \]

Problem 8096

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

Problem 8097

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

Problem 8098

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

Problem 8099

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (11 x^{2}+5\right ) y^{\prime }+24 y x^{2}=0} \]
program solution	\[ y = \frac {c_{1}}{x^{4}}+\frac {c_{2} \left (-2 x^{2}-1\right )}{4 x^{4} \left (x^{2}+1\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (2 x^{2}+1\right )}{\left (x^{2}+1\right )^{2} x^{4}}+\frac {c_{2}}{\left (x^{2}+1\right )^{2}} \]

Problem 8100

ODE	\[ \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+8 x y^{\prime }-\left (-x^{2}+35\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}+1\right )^{2}}{x^{\frac {7}{2}}}+\frac {c_{2} \left (\ln \left (x^{2}+1\right ) \left (x^{2}+1\right )^{2}+2 x^{2}+\frac {3}{2}\right )}{2 x^{\frac {7}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{4}+2 x^{2}+1\right )}{x^{\frac {7}{2}}}+\frac {c_{2} \left (\frac {\ln \left (x^{2}+1\right ) x^{4}}{2}+\ln \left (x^{2}+1\right ) x^{2}+x^{2}+\frac {\ln \left (x^{2}+1\right )}{2}+\frac {3}{4}\right )}{x^{\frac {7}{2}}} \]