2.17.84 Problems 8301 to 8400

Problem 8301

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 8302

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

Problem 8303

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 8304

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 8305

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 8306

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 8307

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 8308

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 8309

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 8310

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 8311

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 8312

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 8313

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 8314

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

Problem 8315

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

Problem 8316

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

Problem 8317

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

Problem 8318

ODE	\[ \boxed {y^{\prime \prime }-\frac {20 y}{x^{2}}=0} \]
program solution	\[ y = \frac {c_{1}}{x^{4}}+\frac {c_{2} x^{5}}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{x^{4}}+c_{2} x^{5} \]

Problem 8319

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

Problem 8320

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

Problem 8321

ODE	\[ \boxed {x y^{\prime \prime }-\left (2 x +2\right ) y^{\prime }+\left (2+x \right ) y=0} \]
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 8322

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

Problem 8323

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

Problem 8324

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

Problem 8325

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

Problem 8326

ODE	\[ \boxed {y^{\prime \prime }-\frac {\left (4 x^{6}-8 x^{5}+12 x^{4}+4 x^{3}+7 x^{2}-20 x +4\right ) y}{4 x^{4}}=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}-1\right ) {\mathrm e}^{\frac {x^{3}-2 x^{2}-2}{2 x}}}{x^{\frac {3}{2}}}+\frac {c_{2} \left (x^{2}-1\right ) {\mathrm e}^{\frac {x^{3}-2 x^{2}-2}{2 x}} \left (\int \frac {x^{3} {\mathrm e}^{\frac {-x^{3}+2 x^{2}+2}{x}}}{\left (x^{2}-1\right )^{2}}d x \right )}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\frac {x^{3}-2 x^{2}-2}{2 x}} \left (x^{2}-1\right )}{x^{\frac {3}{2}}}+\frac {c_{2} {\mathrm e}^{\frac {x^{3}-2 x^{2}-2}{2 x}} \left (x^{2}-1\right ) \left (\int \frac {x^{3} {\mathrm e}^{-\frac {x^{3}-2 x^{2}-2}{x}}}{\left (x -1\right )^{2} \left (x +1\right )^{2}}d x \right )}{x^{\frac {3}{2}}} \]

Problem 8327

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

Problem 8328

ODE	\[ \boxed {y^{\prime \prime }-\left (\frac {x^{2}}{4}-\frac {11}{2}\right ) y=0} \]
program solution	\[ y = c_{1} x \left (x^{4}-10 x^{2}+15\right ) {\mathrm e}^{-\frac {x^{2}}{4}}+c_{2} x \left (x^{4}-10 x^{2}+15\right ) {\mathrm e}^{-\frac {x^{2}}{4}} \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{x^{2} \left (x^{4}-10 x^{2}+15\right )^{2}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{2}}{4}} x \left (x^{4}-10 x^{2}+15\right )+c_{2} {\mathrm e}^{-\frac {x^{2}}{4}} x \left (x^{4}-10 x^{2}+15\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}}}{\left (x^{4}-10 x^{2}+15\right )^{2} x^{2}}d x \right ) \]

Problem 8329

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

Problem 8330

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

Problem 8331

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (5 x^{2}+27\right ) y}{36 \left (x^{2}-1\right )^{2}}=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\int \frac {4 x +\sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}+\sqrt {-\frac {2 \left (\left (-x^{2}+\frac {\left (x^{2}-1\right )^{\frac {2}{3}}}{2}+1\right ) \sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}+x^{3}-x \right )}{\sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}}}}{6 x^{2}-6}d x}+c_{2} {\mathrm e}^{\frac {\left (\int \frac {4 x +\sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}+\sqrt {-\frac {2 \left (\left (-x^{2}+\frac {\left (x^{2}-1\right )^{\frac {2}{3}}}{2}+1\right ) \sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}+x^{3}-x \right )}{\sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}}}}{x^{2}-1}d x \right )}{6}} \left (\int {\mathrm e}^{-\frac {\left (\int \frac {4 x +\sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}+\sqrt {-\frac {2 \left (\left (-x^{2}+\frac {\left (x^{2}-1\right )^{\frac {2}{3}}}{2}+1\right ) \sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}+x^{3}-x \right )}{\sqrt {x^{2}-1+\left (x^{2}-1\right )^{\frac {2}{3}}}}}}{x^{2}-1}d x \right )}{3}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{2}-1\right )^{\frac {1}{3}} {\mathrm e}^{\int \operatorname {RootOf}\left (-1+\left (432 x^{4}-864 x^{2}+432\right ) \textit {\_Z}^{4}+\left (-72 x^{2}+72\right ) \textit {\_Z}^{2}+16 x \textit {\_Z} , \operatorname {index} =1\right )d x}+c_{2} \left (x^{2}-1\right )^{\frac {1}{3}} {\mathrm e}^{\int \operatorname {RootOf}\left (-1+\left (432 x^{4}-864 x^{2}+432\right ) \textit {\_Z}^{4}+\left (-72 x^{2}+72\right ) \textit {\_Z}^{2}+16 x \textit {\_Z} , \operatorname {index} =2\right )d x} \]

Problem 8332

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

Problem 8333

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 8334

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

Problem 8335

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 8336

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

Problem 8337

ODE	\[ \boxed {\left (x -2\right )^{2} y^{\prime \prime }-\left (x -2\right ) y^{\prime }-3 y=0} \]
program solution	\[ y = \frac {c_{1}}{x -2}+\frac {c_{2} \left (x -2\right )^{3}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{x -2}+\frac {c_{2} \left (x^{4}-8 x^{3}+24 x^{2}-32 x \right )}{x -2} \]

Problem 8338

ODE	\[ \boxed {y^{\prime }=\frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}}} \]
program solution	\[ y = \int \frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}}d x +c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \int \frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}}d x +c_{1} \]

Problem 8339

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

Problem 8340

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

Problem 8341

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

Problem 8342

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

Problem 8343

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

Problem 8344

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

Problem 8345

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

Problem 8346

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

Problem 8347

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

Problem 8348

ODE	\[ \boxed {y^{\prime }+f \left (x \right ) y=g \left (x \right )} \]
program solution	\[ \int _{}^{x}\left (f \left (\textit {\_a} \right ) y-g \left (\textit {\_a} \right )\right ) {\mathrm e}^{\int f \left (\textit {\_a} \right )d \textit {\_a}}d \textit {\_a} +\left (-{\mathrm e}^{\int _{}^{x}f \left (\textit {\_a} \right )d \textit {\_a}}+{\mathrm e}^{\int f \left (x \right )d x}\right ) y = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\int g \left (x \right ) {\mathrm e}^{\int f \left (x \right )d x}d x +c_{1} \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x \right )} \]

Problem 8349

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

Problem 8350

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

Problem 8351

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

Problem 8352

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

Problem 8353

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

Problem 8354

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

Problem 8355

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

Problem 8356

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

Problem 8357

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

Problem 8358

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

Problem 8359

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

Problem 8360

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

Problem 8361

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

Problem 8362

ODE	\[ \boxed {y^{\prime }+a y^{2}=b \,x^{2 \nu }+c \,x^{-1+\nu }} \]
program solution	\[ y = \frac {c_{3} \left (-\sqrt {a}\, \sqrt {b}\, c +\left (\nu +2\right ) b \right ) \operatorname {WhittakerM}\left (-\frac {\left (-2 \nu -2\right ) \sqrt {b}+\sqrt {a}\, c}{\sqrt {b}\, \left (2+2 \nu \right )}, \frac {1}{2+2 \nu }, \frac {2 \sqrt {b}\, \sqrt {a}\, x^{1+\nu }}{1+\nu }\right )-2 b \left (1+\nu \right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 \nu -2\right ) \sqrt {b}+\sqrt {a}\, c}{\sqrt {b}\, \left (2+2 \nu \right )}, \frac {1}{2+2 \nu }, \frac {2 \sqrt {b}\, \sqrt {a}\, x^{1+\nu }}{1+\nu }\right )+2 \left (x^{1+\nu } \sqrt {a}\, b^{\frac {3}{2}}+\frac {\sqrt {a}\, \sqrt {b}\, c}{2}-\frac {\nu b}{2}\right ) \left (\operatorname {WhittakerM}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2+2 \nu \right )}, \frac {1}{2+2 \nu }, \frac {2 \sqrt {b}\, \sqrt {a}\, x^{1+\nu }}{1+\nu }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2+2 \nu \right )}, \frac {1}{2+2 \nu }, \frac {2 \sqrt {b}\, \sqrt {a}\, x^{1+\nu }}{1+\nu }\right )\right )}{2 a b x \left (\operatorname {WhittakerM}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2+2 \nu \right )}, \frac {1}{2+2 \nu }, \frac {2 \sqrt {b}\, \sqrt {a}\, x^{1+\nu }}{1+\nu }\right ) c_{3} +\operatorname {WhittakerW}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2+2 \nu \right )}, \frac {1}{2+2 \nu }, \frac {2 \sqrt {b}\, \sqrt {a}\, x^{1+\nu }}{1+\nu }\right )\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (\frac {\nu }{2}+1\right ) \sqrt {b}-\frac {\sqrt {a}\, c}{2}\right ) \operatorname {WhittakerM}\left (-\frac {\left (-2 \nu -2\right ) \sqrt {b}+\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )-c_{1} \sqrt {b}\, \left (\nu +1\right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 \nu -2\right ) \sqrt {b}+\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )+\left (\operatorname {WhittakerW}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )\right ) \left (x^{\nu +1} b \sqrt {a}+\frac {\sqrt {a}\, c}{2}-\frac {\sqrt {b}\, \nu }{2}\right )}{\sqrt {b}\, \left (\operatorname {WhittakerW}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {\sqrt {a}\, c}{\sqrt {b}\, \left (2 \nu +2\right )}, \frac {1}{2 \nu +2}, \frac {2 \sqrt {a}\, \sqrt {b}\, x^{\nu +1}}{\nu +1}\right )\right ) a x} \]

Problem 8363

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

Problem 8364

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

Problem 8365

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

Problem 8366

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

Problem 8367

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

Problem 8368

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

Problem 8369

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

Problem 8370

ODE	\[ \boxed {y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}=-\frac {g^{\prime }\left (x \right )}{f \left (x \right )}} \]
program solution	\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) g \left (x \right ) f \left (x \right ) f^{\prime }\left (x \right )-f \left (x \right ) g \left (x \right ) f^{\prime \prime }\left (x \right ) \textit {\_Y}^{\prime }\left (x \right )+f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \left (\textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )-\textit {\_Y} \left (x \right ) f^{\prime }\left (x \right )\right )}{g \left (x \right ) f^{\prime }\left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) g \left (x \right )}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) g \left (x \right ) f \left (x \right ) f^{\prime }\left (x \right )-f \left (x \right ) g \left (x \right ) f^{\prime \prime }\left (x \right ) \textit {\_Y}^{\prime }\left (x \right )+f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \left (\textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )-\textit {\_Y} \left (x \right ) f^{\prime }\left (x \right )\right )}{g \left (x \right ) f^{\prime }\left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-g \left (x \right ) f \left (x \right ) \left (\int \frac {\frac {d}{d x}f \left (x \right )}{g \left (x \right ) f \left (x \right )^{2}}d x \right )-g \left (x \right ) f \left (x \right ) c_{1} -1}{f \left (x \right )^{2} \left (\int \frac {\frac {d}{d x}f \left (x \right )}{g \left (x \right ) f \left (x \right )^{2}}d x +c_{1} \right )} \]

Problem 8371

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

Problem 8372

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

Problem 8373

ODE	\[ \boxed {y^{\prime }+y^{3}+a y^{2} x=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {2 a}{a^{2} x^{2}+2 \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\textit {\_Z} \right ) 2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} c_{1} x +2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} x \operatorname {AiryAi}\left (\textit {\_Z} \right )+2 \operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_{1} +2 \operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right ) 2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}}} \]

Problem 8374

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

Problem 8375

ODE	\[ \boxed {y^{\prime }-a y^{3}=\frac {b}{x^{\frac {3}{2}}}} \]
program solution	\[ -\frac {\ln \left (x \right )}{2} = \int _{}^{y \sqrt {x}}-\frac {1}{2 \textit {\_a}^{3} a +\textit {\_a} +2 b}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 a \,\textit {\_a}^{3}+\textit {\_a} +2 b}d \textit {\_a} \right )\right )}{\sqrt {x}} \]

Problem 8376

ODE	\[ \boxed {y^{\prime }-\operatorname {a3} y^{3}-\operatorname {a2} y^{2}-\operatorname {a1} y=\operatorname {a0}} \]
program solution	\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3} \operatorname {a3} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a} = x +c_{1} \] Verified OK.
Maple solution	\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a}^{3} \operatorname {a3} +\textit {\_a}^{2} \operatorname {a2} +\textit {\_a} \operatorname {a1} +\operatorname {a0}}d \textit {\_a} \right )+c_{1} = 0 \]

Problem 8377

ODE	\[ \boxed {y^{\prime }+3 a y^{3}+6 a y^{2} x=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {1}{3 a \,x^{2}+\operatorname {RootOf}\left (3^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} \operatorname {AiryBi}\left (\textit {\_Z} \right ) c_{1} x +3^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x \operatorname {AiryAi}\left (\textit {\_Z} \right )+\operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_{1} +\operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right ) 3^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}}} \]

Problem 8378

ODE	\[ \boxed {y^{\prime }+a x y^{3}+b y^{2}=0} \]
program solution	\[ \frac {2 \ln \left (3 \left (y-\frac {b}{3 a x}\right ) a x -b \right ) \sqrt {b^{2}+4 a}-\ln \left (9 a^{2} x^{2} \left (y-\frac {b}{3 a x}\right )^{2}+\left (3 b \left (y-\frac {b}{3 a x}\right ) x -9\right ) a -2 b^{2}\right ) \sqrt {b^{2}+4 a}+2 b \,\operatorname {arctanh}\left (\frac {6 \left (y-\frac {b}{3 a x}\right ) a x +b}{3 \sqrt {b^{2}+4 a}}\right )}{2 \sqrt {b^{2}+4 a}} = \ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{\operatorname {RootOf}\left (2 \sqrt {b^{2}+4 a}\, b \,\operatorname {arctanh}\left (\frac {2 a \,{\mathrm e}^{\textit {\_Z}}+b}{\sqrt {b^{2}+4 a}}\right )-\ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) b^{2}+2 c_{1} b^{2}+2 \textit {\_Z} \,b^{2}-4 \ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) a +8 c_{1} a +8 a \textit {\_Z} \right )}}{x} \]

Problem 8379

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

Problem 8380

ODE	\[ \boxed {y^{\prime }+\left (4 a^{2} x +3 a \,x^{2}+b \right ) y^{3}+3 x y^{2}=0} \]
program solution
Maple solution	\[ \frac {a \sqrt {3}\, \left (\operatorname {BesselI}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y \left (x \right )-2 a}{y \left (x \right ) a^{3}}}}{2}\right ) c_{1} -\operatorname {BesselK}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y \left (x \right )-2 a}{y \left (x \right ) a^{3}}}}{2}\right )\right ) \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y \left (x \right )-2 a}{y \left (x \right ) a^{3}}}-\left (c_{1} \operatorname {BesselI}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y \left (x \right )-2 a}{y \left (x \right ) a^{3}}}}{2}\right )+\operatorname {BesselK}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y \left (x \right )-2 a}{y \left (x \right ) a^{3}}}}{2}\right )\right ) \left (a \sqrt {\frac {4 a^{3}-3 b}{a^{3}}}-2 a -3 x \right )}{\operatorname {BesselI}\left (1+\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y \left (x \right )-2 a}{y \left (x \right ) a^{3}}}}{2}\right ) \sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y \left (x \right )-2 a}{y \left (x \right ) a^{3}}}\, a -\operatorname {BesselI}\left (\frac {\sqrt {\frac {4 a^{3}-3 b}{a^{3}}}}{2}, -\frac {\sqrt {3}\, \sqrt {\frac {\left (4 a^{2} x +3 a \,x^{2}+b \right ) y \left (x \right )-2 a}{y \left (x \right ) a^{3}}}}{2}\right ) \left (a \sqrt {\frac {4 a^{3}-3 b}{a^{3}}}-2 a -3 x \right )} = 0 \]

Problem 8381

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

Problem 8382

ODE	\[ \boxed {y^{\prime }+2 \left (a^{2} x^{3}-x \,b^{2}\right ) y^{3}+3 b y^{2}=0} \]
program solution
Maple solution	\[ c_{1} +\frac {\left (\frac {a^{2} y \left (x \right )^{2} x^{4}-y \left (x \right )^{2} b^{2} x^{2}+2 b x y \left (x \right )-1}{\left (b x y \left (x \right )-1\right )^{2}}\right )^{\frac {1}{4}} a x}{\sqrt {\frac {a \,x^{2} y \left (x \right )}{b x y \left (x \right )-1}}\, b \left (b x y \left (x \right )-1\right )}-\left (\int _{}^{\frac {a \,x^{2} y \left (x \right )}{b x y \left (x \right )-1}}\frac {\left (\textit {\_a}^{2}-1\right )^{\frac {1}{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} \right ) = 0 \]

Problem 8383

ODE	\[ \boxed {y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y=x^{-2 a}-a \,x^{-a -1}} \]
program solution	\[ \frac {-32 \left (y+x^{-a}\right )^{2} {\mathrm e}^{\frac {i \pi +2 x^{1-a}}{-1+a}} \left (\left (x -\frac {x^{-1+2 a}}{4}\right ) 2^{\frac {-3 a +5}{-1+a}}+\frac {x^{-1+2 a} 4^{\frac {1}{-1+a}}}{32}\right ) \left (-1+a \right )^{\frac {-2+a}{-1+a}} \operatorname {WhittakerM}\left (-\frac {1}{-1+a}, \frac {a -3}{2 a -2}, -\frac {4 x^{1-a}}{-1+a}\right )-\left (a -3\right ) \left (\left (4 \left (y+x^{-a}\right )^{2} x^{2}+2 x^{a +1} \left (y+x^{-a}\right )^{2}+a +1\right ) {\mathrm e}^{\frac {4 x^{1-a}}{-1+a}}+2 c_{1} \left (y+x^{-a}\right )^{2} \left (a +1\right )\right )}{2 \left (a +1\right ) \left (a -3\right ) \left (y+x^{-a}\right )^{2}} = 0 \] Warning, solution could not be verified
Maple solution	\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

Problem 8384

ODE	\[ \boxed {y^{\prime }-a \left (x^{n}-x \right ) y^{3}-y^{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8385

ODE	\[ \boxed {y^{\prime }-\left (a \,x^{n}+b x \right ) y^{3}-c y^{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8386

ODE	\[ \boxed {y^{\prime }+a \phi ^{\prime }\left (x \right ) y^{3}+6 a \phi \left (x \right ) y^{2}+\frac {\left (1+2 a \right ) y \phi ^{\prime \prime }\left (x \right )}{\phi ^{\prime }\left (x \right )}=-2 a -2} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8387

ODE	\[ \boxed {y^{\prime }-f_{3} \left (x \right ) y^{3}-f_{2} \left (x \right ) y^{2}-f_{1} \left (x \right ) y=f_{0} \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8388

ODE	\[ \boxed {y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {2 \left (f \left (x \right )-g \left (x \right )\right ) \left (a +\frac {b}{2}\right ) {\mathrm e}^{\operatorname {RootOf}\left (-2 a^{3} b \left (\int g \left (x \right ) f \left (x \right ) h \left (x \right )d x \right )-2 a^{2} b^{2} \left (\int g \left (x \right ) f \left (x \right ) h \left (x \right )d x \right )-2 a \,b^{3} \left (\int g \left (x \right ) f \left (x \right ) h \left (x \right )d x \right )+a^{3} b \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a^{2} b^{2} \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a \,b^{3} \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a^{3} b \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )+a^{2} b^{2} \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )+a \,b^{3} \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )-2 a^{3} b \ln \left (\frac {-9 a^{3}-18 a^{2} b -18 a \,b^{2}-9 b^{3}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a +2 b}\right )-2 a^{2} b^{2} \ln \left (\frac {-9 a^{3}-18 a^{2} b -18 a \,b^{2}-9 b^{3}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a +2 b}\right )-a \,b^{3} \ln \left (\frac {-9 a^{3}-18 a^{2} b -18 a \,b^{2}-9 b^{3}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a +2 b}\right )+3 \ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{3} b +4 \ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{2} b^{2}+3 \ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a \,b^{3}-\textit {\_Z} \,a^{3} b -2 \textit {\_Z} \,a^{2} b^{2}-2 \textit {\_Z} a \,b^{3}+3 c_{1} a^{3} b +6 c_{1} a^{2} b^{2}+3 c_{1} a \,b^{3}-a^{4} \ln \left (\frac {-9 a^{3}-18 a^{2} b -18 a \,b^{2}-9 b^{3}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a +2 b}\right )+\ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{4}+\ln \left (\frac {-9 a^{3}-9 a^{2} b -9 a \,b^{2}+2 a \,{\mathrm e}^{\textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) b^{4}-\textit {\_Z} \,b^{4}\right )}+9 \left (a +b \right ) \left (a^{2}+a b +b^{2}\right ) g \left (x \right )}{9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}} \]

Problem 8389

ODE	\[ \boxed {y^{\prime }-a y^{n}=b \,x^{\frac {n}{-n +1}}} \]
program solution	\[ -\frac {\ln \left (x \right )}{n -1}+\int _{}^{y x^{\frac {1}{n -1}}}\frac {1}{\textit {\_a}^{n} \left (n -1\right ) a +b n -b +\textit {\_a}}d \textit {\_a} -c_{1} = 0 \] Verified OK.
Maple solution	\[ x^{\frac {n}{n -1}} \left (\int _{\textit {\_b}}^{y \left (x \right )}\frac {1}{a \,\textit {\_a}^{n} \left (n -1\right ) x^{\frac {2 n -1}{n -1}}+x^{\frac {n}{n -1}} \textit {\_a} +b x \left (n -1\right )}d \textit {\_a} \right )-c_{1} = 0 \]

Problem 8390

ODE	\[ \boxed {y^{\prime }-f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}=g^{\prime }\left (x \right ) f \left (x \right )} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-f \left (x \right )^{n} \left (\left (\frac {d}{d x}g \left (x \right )\right )^{3} f \left (x \right )^{-n +2} n a \left (a g \left (x \right )+b \right )^{-1-n}\right )^{n} \left (a g \left (x \right )+b \right )^{n} \left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} f \left (x \right )^{n} \left (a g \left (x \right )+b \right )^{n} \left (\left (\frac {d}{d x}g \left (x \right )\right )^{3} f \left (x \right )^{-n +2} n a \left (a g \left (x \right )+b \right )^{-1-n}\right )^{n}-f \left (x \right )^{n} \left (\left (\frac {d}{d x}g \left (x \right )\right )^{3} f \left (x \right )^{-n +2} n a \left (a g \left (x \right )+b \right )^{-1-n}\right )^{n} \left (a g \left (x \right )+b \right )^{n}-\textit {\_a}^{n} \left (\left (\frac {d}{d x}g \left (x \right )\right ) f \left (x \right )^{-n +1} \left (a g \left (x \right )+b \right )^{-n}\right )^{n} \left (\left (\frac {d}{d x}g \left (x \right )\right ) f \left (x \right )\right )^{2 n} n^{n}}d \textit {\_a} \right )-\ln \left (a g \left (x \right )+b \right )+c_{1} \right ) \left (a g \left (x \right )+b \right ) f \left (x \right )}{a} \]

Problem 8391

ODE	\[ \boxed {y^{\prime }-a^{n} f \left (x \right )^{-n +1} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}=g^{\prime }\left (x \right ) f \left (x \right )} \]
program solution
Maple solution	\[ \frac {a y \left (x \right ) \operatorname {LerchPhi}\left (-\left (\frac {a y \left (x \right )}{f \left (x \right )}\right )^{n}, 1, \frac {1}{n}\right )}{n f \left (x \right )}-a g \left (x \right )+c_{1} = 0 \]

Problem 8392

ODE	\[ \boxed {y^{\prime }-f \left (x \right ) y^{n}-g \left (x \right ) y=h \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8393

ODE	\[ \boxed {y^{\prime }-f \left (x \right ) y^{a}-g \left (x \right ) y^{b}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8394

ODE	\[ \boxed {y^{\prime }-\sqrt {{| y|}}=0} \]
program solution	\[ \left \{\begin {array}{cc} -2 \sqrt {-y} & y\le 0 \\ 2 \sqrt {y} & 0
Maple solution	\[ x +2 \left (\left \{\begin {array}{cc} \sqrt {-y \left (x \right )} & y \left (x \right )\le 0 \\ -\sqrt {y \left (x \right )} & 0

Problem 8395

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

Problem 8396

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

Problem 8397

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

Problem 8398

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

Problem 8399

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

Problem 8400

ODE	\[ \boxed {y^{\prime }-\frac {y^{2}+1}{{| y+\sqrt {y+1}|} \left (x +1\right )^{\frac {3}{2}}}=0} \]
program solution	\[ \text {Expression too large to display} \] Warning, solution could not be verified
Maple solution	\[ -\frac {2}{\sqrt {x +1}}-\left (\int _{}^{y \left (x \right )}\frac {{| \textit {\_a} +\sqrt {\textit {\_a} +1}|}}{\textit {\_a}^{2}+1}d \textit {\_a} \right )+c_{1} = 0 \]