2.17.83 Problems 8201 to 8300

Problem 8201

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

Problem 8202

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 8203

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

Problem 8204

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

Problem 8205

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

Problem 8206

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

Problem 8207

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

Problem 8208

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 8209

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 8210

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 8211

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 8212

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

Problem 8213

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

Problem 8214

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

Problem 8215

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 8216

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

Problem 8217

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

Problem 8218

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 8219

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

Problem 8220

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

Problem 8221

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

Problem 8222

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

Problem 8223

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

Problem 8224

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

Problem 8225

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

Problem 8226

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

Problem 8227

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 8228

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 8229

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+12 y=0} \]
program solution	\[ y = \frac {c_{1} \left (5 x^{3}-3 x \right ) \sqrt {x^{2}-1}}{5 \sqrt {x -1}\, \sqrt {x +1}}+\frac {5 c_{2} \left (15 \ln \left (x -1\right ) x^{3}-15 \ln \left (x +1\right ) x^{3}-9 \ln \left (x -1\right ) x +9 \ln \left (x +1\right ) x +30 x^{2}-8\right ) \sqrt {x^{2}-1}}{24 \sqrt {x +1}\, \sqrt {x -1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (-\frac {5}{3} x^{3}+x \right )+c_{2} \left (-\frac {5 \ln \left (x +1\right ) x^{3}}{24}+\frac {5 \ln \left (x -1\right ) x^{3}}{24}+\frac {\ln \left (x +1\right ) x}{8}-\frac {\ln \left (x -1\right ) x}{8}+\frac {5 x^{2}}{12}-\frac {1}{9}\right ) \]

Problem 8230

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

Problem 8231

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

Problem 8232

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 8233

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 8234

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 8235

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 8236

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 8237

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 8238

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 8239

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 8240

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 8241

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 8242

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 8243

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 8244

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 8245

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 8246

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 8247

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 8248

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 8249

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 8250

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 8251

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 8252

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 8253

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 8254

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 8255

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 8256

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 8257

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 8258

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 8259

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 8260

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 8261

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 8262

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 8263

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 8264

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 8265

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 8266

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 8267

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 8268

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 8269

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 8270

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 8271

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 8272

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 8273

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 8274

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 8275

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 8276

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 8277

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 8278

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 8279

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 8280

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 8281

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 8282

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

Problem 8283

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 8284

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 8285

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 8286

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 8287

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 8288

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 8289

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 8290

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 8291

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 8292

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 8293

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 8294

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 8295

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 8296

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 8297

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 8298

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 8299

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 8300

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	\[ y \left (x \right ) = \frac {c_{1} \left (9 x^{2}-6 x +2\right )}{x^{2}}+\frac {c_{2} {\mathrm e}^{-3 x}}{x^{2}} \]