2.17.82 Problems 8101 to 8200

Problem 8101

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) y=0} \]
program solution	\[ y = \frac {c_{1}}{x \left (x^{2}+1\right )^{2}}+\frac {c_{2} x^{7} \left (4 x^{2}+5\right )}{40 \left (x^{2}+1\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\left (x^{2}+1\right )^{2} x}+\frac {c_{2} \left (x^{10}+\frac {5}{4} x^{8}\right )}{\left (x^{2}+1\right )^{2} x} \]

Problem 8102

ODE	\[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}+8\right ) \left (x^{2}+1\right )^{\frac {5}{2}}}{x^{7}}+\frac {c_{2} \left (-35 x^{6}-140 x^{4}-168 x^{2}-64\right )}{35 x^{7}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (35 x^{6}+140 x^{4}+168 x^{2}+64\right )}{x^{7}}+\frac {c_{2} \left (x^{2}+1\right )^{\frac {5}{2}} \left (x^{2}+8\right )}{x^{7}} \]

Problem 8103

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

Problem 8104

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

Problem 8105

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

Problem 8106

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

Problem 8107

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

Problem 8108

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

Problem 8109

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

Problem 8110

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

Problem 8111

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

Problem 8112

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

Problem 8113

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

Problem 8114

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

Problem 8115

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

Problem 8116

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

Problem 8117

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

Problem 8118

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

Problem 8119

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

Problem 8120

ODE	\[ \boxed {t y^{\prime \prime }-\left (4+t \right ) y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} \left (t^{2}+6 t +12\right )+c_{2} \left (t^{2}-6 t +12\right ) {\mathrm e}^{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \left (t^{2}+6 t +12\right )+c_{2} {\mathrm e}^{t} \left (t^{2}-6 t +12\right ) \]

Problem 8121

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

Problem 8122

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

Problem 8123

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

Problem 8124

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

Problem 8125

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

Problem 8126

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

Problem 8127

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

Problem 8128

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

Problem 8129

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

Problem 8130

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

Problem 8131

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

Problem 8132

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

Problem 8133

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

Problem 8134

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-1\right ) y=0} \]
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 8135

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

Problem 8136

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

Problem 8137

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

Problem 8138

ODE	\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-3 y x=0} \]
program solution	\[ y = c_{1} x \,{\mathrm e}^{\frac {x^{3}}{3}}+c_{2} x \,{\mathrm e}^{\frac {x^{3}}{3}} \left (\int \frac {{\mathrm e}^{-\frac {x^{3}}{3}}}{x^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x^{3}}{3}} x -\frac {9 c_{2} {\mathrm e}^{\frac {x^{3}}{3}} 3^{\frac {2}{3}} {\mathrm e}^{-\frac {x^{3}}{6}} \left (x^{6} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right )+5 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right ) x^{3}+10 \operatorname {WhittakerM}\left (\frac {4}{3}, \frac {5}{6}, \frac {x^{3}}{3}\right )\right )}{10 x^{3} \left (x^{3}\right )^{\frac {1}{3}}} \]

Problem 8139

ODE	\[ \boxed {\left (-4 x^{2}+1\right ) y^{\prime \prime }-20 x y^{\prime }-16 y=0} \]
program solution	\[ y = \frac {c_{1} x \sqrt {2}}{\left (4 x^{2}-1\right )^{\frac {3}{2}}}-\frac {c_{2} \sqrt {2}\, \left (-2 \ln \left (2 x +\sqrt {4 x^{2}-1}\right ) x +\sqrt {4 x^{2}-1}\right )}{2 \left (4 x^{2}-1\right )^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x}{\left (4 x^{2}-1\right )^{\frac {3}{2}}}+\frac {c_{2} \left (2 \ln \left (2 x +\sqrt {4 x^{2}-1}\right ) x -\sqrt {4 x^{2}-1}\right )}{\left (4 x^{2}-1\right )^{\frac {3}{2}}} \]

Problem 8140

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

Problem 8141

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

Problem 8142

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 8143

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

Problem 8144

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

Problem 8145

ODE	\[ \boxed {4 x y^{\prime \prime }-x y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} \left (x -8\right ) x +c_{2} \left (\frac {\left (-x^{2}+8 x \right ) \operatorname {expIntegral}_{1}\left (-\frac {x}{4}\right )}{128}-\frac {{\mathrm e}^{\frac {x}{4}} \left (x -4\right )}{32}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{2}-8 x \right )+c_{2} \left (\frac {\operatorname {expIntegral}_{1}\left (-\frac {x}{4}\right ) x^{2}}{128}+\frac {{\mathrm e}^{\frac {x}{4}} x}{32}-\frac {\operatorname {expIntegral}_{1}\left (-\frac {x}{4}\right ) x}{16}-\frac {{\mathrm e}^{\frac {x}{4}}}{8}\right ) \]

Problem 8146

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

Problem 8147

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

Problem 8148

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-x \left (2 x +1\right ) y^{\prime }+2 \left (4 x -1\right ) y=0} \]
program solution	\[ y = c_{1} \left (x^{4}-9 x^{3}+\frac {63}{4} x^{2}\right )+16 c_{2} \left (x^{2}-9 x +\frac {63}{4}\right ) x^{2} \left (\int \frac {{\mathrm e}^{x}}{x^{\frac {7}{2}} \left (4 x^{2}-36 x +63\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (4 x^{2}-36 x +63\right )+c_{2} x^{2} \left (4 x^{2}-36 x +63\right ) \left (\int \frac {{\mathrm e}^{x}}{\left (4 x^{2}-36 x +63\right )^{2} x^{\frac {7}{2}}}d x \right ) \]

Problem 8149

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

Problem 8150

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

Problem 8151

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

Problem 8152

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

Problem 8153

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (2 \sqrt {5}-1\right ) x y^{\prime }+\left (\frac {19}{4}-3 x^{2}\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\sqrt {3}\, x} x^{\sqrt {5}-\frac {1}{2}}+\frac {c_{2} \sqrt {3}\, x^{\sqrt {5}-\frac {1}{2}} {\mathrm e}^{\sqrt {3}\, x}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{\sqrt {5}} \sinh \left (\sqrt {3}\, x \right )}{\sqrt {x}}+\frac {c_{2} x^{\sqrt {5}} \cosh \left (\sqrt {3}\, x \right )}{\sqrt {x}} \]

Problem 8154

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

Problem 8155

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

Problem 8156

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

Problem 8157

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

Problem 8158

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

Problem 8159

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

Problem 8160

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

Problem 8161

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

Problem 8162

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

Problem 8163

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

Problem 8164

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

Problem 8165

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

Problem 8166

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

Problem 8167

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

Problem 8168

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

Problem 8169

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

Problem 8170

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

Problem 8171

ODE	\[ \boxed {2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y=0} \]
program solution	\[ y = \frac {c_{1} \left (10 x +1\right )}{10 x^{\frac {3}{2}}}+\frac {c_{2} \left (100 x +10\right ) \left (\int \frac {\sqrt {x}\, {\mathrm e}^{5 x}}{\left (10 x +1\right )^{2}}d x \right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (10 x +1\right )}{x^{\frac {3}{2}}}+\frac {c_{2} \left (10 x +1\right ) \left (\int \frac {\sqrt {x}\, {\mathrm e}^{5 x}}{\left (10 x +1\right )^{2}}d x \right )}{x^{\frac {3}{2}}} \]

Problem 8172

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y=0} \]
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 8173

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

Problem 8174

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

Problem 8175

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

Problem 8176

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

Problem 8177

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

Problem 8178

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

Problem 8179

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

Problem 8180

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (-9+x \right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}-8 x +20\right )}{x^{3}}-\frac {c_{2} \left (x^{3}+9 x^{2}+36 x +60\right ) {\mathrm e}^{-x}}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{2}-8 x +20\right )}{x^{3}}+\frac {c_{2} {\mathrm e}^{-x} \left (x^{3}+9 x^{2}+36 x +60\right )}{x^{3}} \]

Problem 8181

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

Problem 8182

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

Problem 8183

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x}=0} \]
program solution	\[ y = \frac {c_{1} \left (5 x +2\right )}{5 \sqrt {x}}+\frac {c_{2} \left (25 x +10\right ) \left (\int \frac {x^{\frac {5}{2}} {\mathrm e}^{-\frac {1}{x}}}{\left (5 x +2\right )^{2}}d x \right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (5 x +2\right )}{\sqrt {x}}+\frac {c_{2} \left (5 x +2\right ) \left (\int \frac {x^{\frac {5}{2}} {\mathrm e}^{-\frac {1}{x}}}{\left (5 x +2\right )^{2}}d x \right )}{\sqrt {x}} \]

Problem 8184

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

Problem 8185

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

Problem 8186

ODE	\[ \boxed {2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y=0} \]
program solution	\[ y = \frac {c_{1} \left (x +1\right ) \sqrt {x}}{\left (x -1\right )^{4}}+\frac {c_{2} \left (2 x^{2}+12 x +2\right )}{\left (x -1\right )^{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (x^{2}+6 x +1\right )}{\left (x -1\right )^{4}}+\frac {c_{2} \sqrt {x}\, \left (x +1\right )}{\left (x -1\right )^{4}} \]

Problem 8187

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9}=0} \]
program solution	\[ y = \frac {c_{1} \left (6 x -1\right ) x^{\frac {1}{6}} \left (x -1\right )^{\frac {5}{6}}}{6 \left (x \left (x -1\right )\right )^{\frac {1}{6}}}-\frac {9 c_{2} \left (6 x -5\right ) \left (x -1\right )^{\frac {1}{6}} x^{\frac {5}{6}}}{5 \left (x \left (x -1\right )\right )^{\frac {1}{6}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (6 x -5\right ) x^{\frac {2}{3}}+c_{2} \left (6 x -1\right ) \left (x -1\right )^{\frac {2}{3}} \]

Problem 8188

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

Problem 8189

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

Problem 8190

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

Problem 8191

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

Problem 8192

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

Problem 8193

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

Problem 8194

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

Problem 8195

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

Problem 8196

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

Problem 8197

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

Problem 8198

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

Problem 8199

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

Problem 8200

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