2.17.79 Problems 7801 to 7900

Problem 7801

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

Problem 7802

ODE	\[ \boxed {\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y=0} \]
program solution	\[ y = \frac {c_{1} \left (3 x -4\right ) \sqrt {x^{2}-2 x +10}\, {\mathrm e}^{-\frac {\arctan \left (\frac {x}{3}-\frac {1}{3}\right )}{3}}}{3}+c_{2} \left (-\frac {9}{410} x^{2}+\frac {6}{205} x -\frac {9}{82}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{2}-\frac {4}{3} x +5\right )+c_{2} \left (3 x -4\right ) \sqrt {x^{2}-2 x +10}\, \left (\frac {-x +1+3 i}{x -1+3 i}\right )^{\frac {i}{6}} \]

Problem 7803

ODE	\[ \boxed {\left (x^{2}-2 x +10\right ) y^{\prime \prime }+x y^{\prime }-4 y=0} \]
program solution	\[ y = \frac {c_{1} \left (3 x -4\right ) \sqrt {x^{2}-2 x +10}\, {\mathrm e}^{-\frac {\arctan \left (\frac {x}{3}-\frac {1}{3}\right )}{3}}}{3}+c_{2} \left (-\frac {9}{410} x^{2}+\frac {6}{205} x -\frac {9}{82}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{2}-\frac {4}{3} x +5\right )+c_{2} \left (3 x -4\right ) \sqrt {x^{2}-2 x +10}\, \left (\frac {-x +1+3 i}{x -1+3 i}\right )^{\frac {i}{6}} \]

Problem 7804

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

Problem 7805

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

Problem 7806

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

Problem 7807

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

Problem 7808

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

Problem 7809

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

Problem 7810

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

Problem 7811

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

Problem 7812

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

Problem 7813

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

Problem 7814

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

Problem 7815

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 7816

ODE	\[ \boxed {x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y=0} \]
program solution	\[ y = c_{1} \left (x^{3}+12 x^{2}+60 x +120\right ) {\mathrm e}^{-x}+c_{2} \left (x^{3}-12 x^{2}+60 x -120\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{3}-12 x^{2}+60 x -120\right )+c_{2} {\mathrm e}^{-x} \left (x^{3}+12 x^{2}+60 x +120\right ) \]

Problem 7817

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

Problem 7818

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y=0} \]
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 7819

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \cos \left (6 x \right )}{\sqrt {x}}+\frac {c_{2} \sin \left (6 x \right )}{6 \sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (6 x \right )}{\sqrt {x}}+\frac {c_{2} \cos \left (6 x \right )}{\sqrt {x}} \]

Problem 7820

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

Problem 7821

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

Problem 7822

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

Problem 7823

ODE	\[ \boxed {16 x^{2} y^{\prime \prime }+32 x y^{\prime }+\left (x^{4}-12\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{-\frac {i x^{2}}{8}}}{x^{\frac {3}{2}}}-\frac {2 i c_{2} {\mathrm e}^{\frac {i x^{2}}{8}}}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (\frac {x^{2}}{8}\right )}{x^{\frac {3}{2}}}+\frac {c_{2} \cos \left (\frac {x^{2}}{8}\right )}{x^{\frac {3}{2}}} \]

Problem 7824

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

Problem 7825

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

Problem 7826

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

Problem 7827

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 7828

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

Problem 7829

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+30 y=0} \]
program solution	\[ y = \frac {c_{1} \left (63 x^{5}-70 x^{3}+15 x \right ) \sqrt {x^{2}-1}}{63 \sqrt {x -1}\, \sqrt {x +1}}+\frac {3969 c_{2} \sqrt {x^{2}-1}\, \left (\left (x^{5}-\frac {10}{9} x^{3}+\frac {5}{21} x \right ) \ln \left (x -1\right )+\left (-x^{5}+\frac {10}{9} x^{3}-\frac {5}{21} x \right ) \ln \left (x +1\right )+2 x^{4}-\frac {14 x^{2}}{9}+\frac {128}{945}\right )}{128 \sqrt {x +1}\, \sqrt {x -1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (\frac {21}{5} x^{5}-\frac {14}{3} x^{3}+x \right )+c_{2} \left (-\frac {21 \ln \left (x +1\right ) x^{5}}{640}+\frac {21 \ln \left (x -1\right ) x^{5}}{640}+\frac {7 \ln \left (x +1\right ) x^{3}}{192}-\frac {7 \ln \left (x -1\right ) x^{3}}{192}+\frac {21 x^{4}}{320}-\frac {\ln \left (x +1\right ) x}{128}+\frac {\ln \left (x -1\right ) x}{128}-\frac {49 x^{2}}{960}+\frac {1}{225}\right ) \]

Problem 7830

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 7831

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

Problem 7832

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

Problem 7833

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

Problem 7834

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

Problem 7835

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

Problem 7836

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 7837

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}+2 x \right ) y^{\prime }-2 y=0} \]
program solution	\[ y = c_{1} \left (x +\frac {1}{4}\right )+\frac {c_{2} \sqrt {x -1}\, \sqrt {x}\, \left (4 \sqrt {x \left (x -1\right )}\, x -12 x \ln \left (2 x -1+2 \sqrt {x \left (x -1\right )}\right )+12 x \ln \left (2\right )+26 \sqrt {x \left (x -1\right )}-3 \ln \left (2 x -1+2 \sqrt {x \left (x -1\right )}\right )+3 \ln \left (2\right )\right )}{4 \sqrt {x \left (x -1\right )}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+4 x \right )+c_{2} \left (4 \sqrt {x \left (x -1\right )}\, x -12 \ln \left (x -\frac {1}{2}+\sqrt {x \left (x -1\right )}\right ) x +26 \sqrt {x \left (x -1\right )}-3 \ln \left (x -\frac {1}{2}+\sqrt {x \left (x -1\right )}\right )\right ) \]

Problem 7838

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

Problem 7839

ODE	\[ \boxed {2 \left (t^{2}-5 t +6\right ) y^{\prime \prime }+\left (2 t -3\right ) y^{\prime }-8 y=0} \]
program solution	\[ y = \frac {c_{1} \left (6 t -17\right ) \left (t -2\right )^{\frac {3}{2}}}{6 \sqrt {t -3}}+c_{2} \left (\frac {96}{5} t^{2}-\frac {416}{5} t +\frac {444}{5}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \left (t^{2}-\frac {13}{3} t +\frac {37}{8}\right )+\frac {c_{2} \left (6 t -17\right ) \left (t -2\right )^{\frac {3}{2}}}{\sqrt {t -3}} \]

Problem 7840

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

Problem 7841

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

Problem 7842

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

Problem 7843

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 7844

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

Problem 7845

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

Problem 7846

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

Problem 7847

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

Problem 7848

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 7849

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

Problem 7850

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

Problem 7851

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

Problem 7852

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

Problem 7853

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 7854

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

Problem 7855

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

Problem 7856

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

Problem 7857

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

Problem 7858

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

Problem 7859

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

Problem 7860

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

Problem 7861

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

Problem 7862

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

Problem 7863

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

Problem 7864

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

Problem 7865

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

Problem 7866

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

Problem 7867

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 7868

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

Problem 7869

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

Problem 7870

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

Problem 7871

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

Problem 7872

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 7873

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 7874

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

Problem 7875

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 7876

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

Problem 7877

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 7878

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 x y^{\prime }+4 x^{4} y=0} \]
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	\[ y \left (x \right ) = \frac {c_{1} \sin \left (x^{2}\right )}{x^{2}}+\frac {c_{2} \cos \left (x^{2}\right )}{x^{2}} \]

Problem 7879

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

Problem 7880

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

Problem 7881

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

Problem 7882

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 7883

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

Problem 7884

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

Problem 7885

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

Problem 7886

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

Problem 7887

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

Problem 7888

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

Problem 7889

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 7890

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 7891

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 7892

ODE	\[ \boxed {\left (x^{2}+3\right ) y^{\prime \prime }-7 x y^{\prime }+16 y=0} \]
program solution	\[ y = c_{1} \left (x^{4}-9 x^{2}+\frac {27}{8}\right )+\frac {c_{2} \left (-256 \left (x^{4}-9 x^{2}+\frac {27}{8}\right ) \left (\left (-x^{3}-\frac {3}{2} x \right ) \sqrt {x^{2}+3}+x^{4}+3 x^{2}+\frac {9}{8}\right ) \ln \left (\sqrt {x^{2}+3}-x \right )+\left (-3840 x^{5}-720 x^{3}+4680 x \right ) \sqrt {x^{2}+3}+3840 x^{6}+6480 x^{4}-7920 x^{2}-2025\right )}{\left (-256 x^{3}-384 x \right ) \sqrt {x^{2}+3}+256 x^{4}+768 x^{2}+288} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{4}-9 x^{2}+\frac {27}{8}\right )+c_{2} \left (\frac {\ln \left (\sqrt {x^{2}+3}-x \right ) x^{4}}{64}+\frac {25 \sqrt {x^{2}+3}\, x^{3}}{768}+\frac {25 x^{4}}{768}-\frac {9 \ln \left (\sqrt {x^{2}+3}-x \right ) x^{2}}{64}-\frac {55 \sqrt {x^{2}+3}\, x}{512}-\frac {75 x^{2}}{256}+\frac {27 \ln \left (\sqrt {x^{2}+3}-x \right )}{512}+\frac {225}{2048}\right ) \]

Problem 7893

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

Problem 7894

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

Problem 7895

ODE	\[ \boxed {5 y^{\prime \prime }-2 x y^{\prime }+10 y=0} \]
program solution	\[ y = c_{1} \left (x^{5}-25 x^{3}+\frac {375}{4} x \right )+16 c_{2} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{5}}}{x^{2} \left (4 x^{4}-100 x^{2}+375\right )^{2}}d x \right ) x \left (x^{4}-25 x^{2}+\frac {375}{4}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{5}-25 x^{3}+\frac {375}{4} x \right )+c_{2} \left (x^{5}-25 x^{3}+\frac {375}{4} x \right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{5}}}{\left (4 x^{4}-100 x^{2}+375\right )^{2} x^{2}}d x \right ) \]

Problem 7896

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 7897

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

Problem 7898

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

Problem 7899

ODE	\[ \boxed {\left (x^{2}-6 x +10\right ) y^{\prime \prime }-4 \left (x -3\right ) y^{\prime }+6 y=0} \]
program solution	\[ y = \frac {c_{1} \left (i x -3 i-1\right )^{2} \left (x^{2}-6 x +10\right )}{-x +3+i}+\frac {c_{2} \left (-x^{2}+6 x -10\right ) \left (x^{2}-6 x +\frac {26}{3}\right )}{\left (-x +3-i\right ) \left (-x +3+i\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (\frac {26}{3}+x^{2}-6 x \right )+c_{2} \left (x^{3}-30 x +60\right ) \]

Problem 7900

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