2.17.98 Problems 9701 to 9800

Problem 9701

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

Problem 9702

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}+\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (\operatorname {LegendreP}\left (v , n , x\right ) c_{1} +\operatorname {LegendreQ}\left (v , n , x\right ) c_{2} \right ) \left (x^{2}-1\right )^{a -\frac {n}{2}} \]

Problem 9703

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

Problem 9704

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

Problem 9705

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

Problem 9706

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

Problem 9707

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

Problem 9708

ODE	\[ \boxed {y^{\prime \prime }+\frac {b y}{x^{2} \left (x -a \right )^{2}}=c} \]
program solution	\[ y = -\frac {\left (\cos \left (\frac {\sqrt {b}}{x}\right ) c_{2} -\sin \left (\frac {\sqrt {b}}{x}\right ) c_{1} \right ) x \left (c \left (\int \frac {b \left (c_{1} i+c_{2} \right ) \operatorname {expIntegral}_{1}\left (-\frac {i \sqrt {b}}{x}\right )-\left (c_{1} i-c_{2} \right ) b \,\operatorname {expIntegral}_{1}\left (\frac {i \sqrt {b}}{x}\right )+2 x \left (\left (-c_{2} x +c_{1} \sqrt {b}\right ) \cos \left (\frac {\sqrt {b}}{x}\right )+\sin \left (\frac {\sqrt {b}}{x}\right ) \left (c_{1} x +c_{2} \sqrt {b}\right )\right )}{x^{2} \left (\sin \left (\frac {\sqrt {b}}{x}\right ) c_{1} -\cos \left (\frac {\sqrt {b}}{x}\right ) c_{2} \right )^{2}}d x \right )+4 c_{3} \right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {x \left (-x +a \right )}\, \left (\left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{1} \sqrt {a^{2}-4 b}+\left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} \left (\int \sqrt {x \left (-x +a \right )}\, \left (\frac {x}{-x +a}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \right ) c +\left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{2} \sqrt {a^{2}-4 b}-\left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} \left (\int \sqrt {x \left (-x +a \right )}\, \left (\frac {-x +a}{x}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \right ) c \right )}{\sqrt {a^{2}-4 b}} \]

Problem 9709

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

Problem 9710

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

Problem 9711

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

Problem 9712

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

Problem 9713

ODE	\[ \boxed {y^{\prime \prime }-\frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )}=0} \]
program solution	\[ y = \frac {c_{1} \left (4 x^{2}+4 x +4\right )}{\left (2 x +1\right )^{2}}+\frac {c_{2} \left (-12 \sqrt {3}\, \left (x^{2}+x +1\right ) \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )+16 x^{3}+16 x^{2}+22 x +3\right )}{4 \left (2 x +1\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-36 c_{2} \left (x^{2}+x +1\right ) \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )+16 c_{2} \left (x^{3}+x^{2}+\frac {11}{8} x +\frac {3}{16}\right ) \sqrt {3}+c_{1} \left (x^{2}+x +1\right )}{\left (2 x +1\right )^{2}} \]

Problem 9714

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

Problem 9715

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

Problem 9716

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (-1+3 x \right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {\left (-v \left (v +1\right ) \left (x -1\right )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\left (1-x \right )^{n -\frac {1}{2}} \left (x -1\right )^{-n} x^{\frac {1}{4}} \left (\Gamma \left (v +\frac {1}{2}\right )^{2} c_{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (n -\frac {1}{2}, -v -\frac {1}{2}, \frac {-x -1}{x -1}\right )+\sec \left (\pi v \right ) \operatorname {LegendreP}\left (n -\frac {1}{2}, v +\frac {1}{2}, \frac {-x -1}{x -1}\right ) \pi c_{1} \right )}{\Gamma \left (v +\frac {1}{2}\right )} \]

Problem 9717

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

Problem 9718

ODE	\[ \boxed {y^{\prime \prime }-\frac {\left (7 x^{2} a +5\right ) y^{\prime }}{x \left (x^{2} a +1\right )}+\frac {\left (15 x^{2} a +5\right ) y}{x^{2} \left (x^{2} a +1\right )}=0} \]
program solution	\[ y = \left (c_{1} \left (-\frac {1}{4 x^{4}}-\frac {a}{2 x^{2}}\right )+c_{2} \right ) x^{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{5}+2 a c_{2} x^{3}+x c_{2} \]

Problem 9719

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

Problem 9720

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

Problem 9721

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

Problem 9722

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

Problem 9723

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

Problem 9724

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

Problem 9725

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

Problem 9726

ODE	\[ \boxed {y^{\prime \prime }-\frac {\left (3 x +1\right ) y^{\prime }}{\left (x -1\right ) \left (x +1\right )}+\frac {36 \left (x +1\right )^{2} y}{\left (x -1\right )^{2} \left (3 x +5\right )^{2}}=0} \]
program solution	\[ y = \frac {c_{1} \left (x -1\right )^{\frac {3}{2}} \sqrt {3}\, \sqrt {3 x +5}}{3}+\frac {c_{2} \left (x -1\right )^{\frac {3}{2}} \sqrt {3}\, \sqrt {3 x +5}\, \left (3 \ln \left (x -1\right )+\ln \left (3 x +5\right )\right )}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x -1\right )^{\frac {3}{2}} \sqrt {3 x +5}\, \left (3 \ln \left (x -1\right ) c_{2} +\ln \left (3 x +5\right ) c_{2} +c_{1} \right ) \]

Problem 9727

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

Problem 9728

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

Problem 9729

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

Problem 9730

ODE	\[ \boxed {y^{\prime \prime }+\left (\frac {1-\operatorname {a1} -\operatorname {b1}}{x -\operatorname {c1}}+\frac {1-\operatorname {a2} -\operatorname {b2}}{x -\operatorname {c2}}+\frac {1-\operatorname {a3} -\operatorname {b3}}{x -\operatorname {c3}}\right ) y^{\prime }+\frac {\left (\frac {\operatorname {a1} \operatorname {b1} \left (\operatorname {c1} -\operatorname {c3} \right ) \left (\operatorname {c1} -\operatorname {c2} \right )}{x -\operatorname {c1}}+\frac {\operatorname {a2} \operatorname {b2} \left (\operatorname {c2} -\operatorname {c1} \right ) \left (\operatorname {c2} -\operatorname {c3} \right )}{x -\operatorname {c2}}+\frac {\operatorname {a3} \operatorname {b3} \left (\operatorname {c3} -\operatorname {c2} \right ) \left (\operatorname {c3} -\operatorname {c1} \right )}{x -\operatorname {c3}}\right ) y}{\left (x -\operatorname {c1} \right ) \left (x -\operatorname {c2} \right ) \left (x -\operatorname {c3} \right )}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (x -\operatorname {c2} \right )^{\operatorname {a2}} \left (x -\operatorname {c3} \right )^{\operatorname {b3}} \left (c_{1} \operatorname {HeunG}\left (\frac {\operatorname {c1} -\operatorname {c3}}{\operatorname {c1} -\operatorname {c2}}, \frac {\left (\left (-\operatorname {a3} -2 \operatorname {b1} -\operatorname {b2} +2\right ) \operatorname {c1} +\left (\operatorname {a3} +\operatorname {b1} -1\right ) \operatorname {c2} +\operatorname {c3} \left (\operatorname {b1} +\operatorname {b2} -1\right )\right ) \operatorname {a1} -\left (\operatorname {b1} -1\right ) \left (\operatorname {a2} +\operatorname {b3} \right ) \operatorname {c1} +\left (\operatorname {b1} \operatorname {b3} -\operatorname {a2} \operatorname {b2} +\operatorname {b3} \left (-1+\operatorname {a3} \right )\right ) \operatorname {c2} +\operatorname {c3} \left (\operatorname {b1} \operatorname {a2} +\left (\operatorname {b2} -1\right ) \operatorname {a2} -\operatorname {a3} \operatorname {b3} \right )}{\operatorname {c1} -\operatorname {c2}}, \operatorname {a1} +\operatorname {b3} +\operatorname {a2} , 2-\operatorname {a3} -\operatorname {b1} -\operatorname {b2} , \operatorname {a1} -\operatorname {b1} +1, \operatorname {a2} -\operatorname {b2} +1, \frac {-x +\operatorname {c1}}{\operatorname {c1} -\operatorname {c2}}\right ) \left (x -\operatorname {c1} \right )^{\operatorname {a1}}+c_{2} \operatorname {HeunG}\left (\frac {\operatorname {c1} -\operatorname {c3}}{\operatorname {c1} -\operatorname {c2}}, \frac {\left (\left (-2 \operatorname {a1} -\operatorname {a3} -\operatorname {b2} +2\right ) \operatorname {c1} +\left (\operatorname {a1} +\operatorname {a3} -1\right ) \operatorname {c2} +\operatorname {c3} \left (\operatorname {a1} +\operatorname {b2} -1\right )\right ) \operatorname {b1} -\left (\operatorname {a2} +\operatorname {b3} \right ) \left (\operatorname {a1} -1\right ) \operatorname {c1} +\left (\operatorname {a1} \operatorname {b3} -\operatorname {a2} \operatorname {b2} +\operatorname {b3} \left (-1+\operatorname {a3} \right )\right ) \operatorname {c2} +\operatorname {c3} \left (\operatorname {a1} \operatorname {a2} +\left (\operatorname {b2} -1\right ) \operatorname {a2} -\operatorname {a3} \operatorname {b3} \right )}{\operatorname {c1} -\operatorname {c2}}, \operatorname {b1} +\operatorname {b3} +\operatorname {a2} , 2-\operatorname {a1} -\operatorname {a3} -\operatorname {b2} , -\operatorname {a1} +\operatorname {b1} +1, \operatorname {a2} -\operatorname {b2} +1, \frac {-x +\operatorname {c1}}{\operatorname {c1} -\operatorname {c2}}\right ) \left (x -\operatorname {c1} \right )^{\operatorname {b1}}\right ) \]

Problem 9731

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

Problem 9732

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

Problem 9733

ODE	\[ \boxed {y^{\prime \prime }+\frac {27 x y}{16 \left (x^{3}-1\right )^{2}}=0} \]
program solution	\[ y = \frac {c_{1} \sqrt {2}\, \left (x -1\right )^{\frac {1}{4}} \left (x^{2}+x +1\right )^{\frac {3}{8}} \left (\frac {\left (\sqrt {3}-3 i\right ) \sqrt {\left (x -1\right ) \left (\sqrt {3}+i\right ) \left (\sqrt {3}+2 i x +i\right )}-3 i \sqrt {3}+6 x -3}{2 x +1+i \sqrt {3}}\right )^{\frac {1}{4}}}{{\left (\frac {3+i \left (2 x +1\right ) \sqrt {3}}{\sqrt {3 x^{2}+3 x +3}}\right )}^{\frac {1}{4}}}+\frac {c_{2} \sqrt {2}\, \left (x -1\right )^{\frac {1}{4}} \left (x^{2}+x +1\right )^{\frac {3}{8}} 3^{\frac {7}{8}} \left (\frac {\left (\sqrt {3}-3 i\right ) \sqrt {\left (x -1\right ) \left (\sqrt {3}+i\right ) \left (\sqrt {3}+2 i x +i\right )}-3 i \sqrt {3}+6 x -3}{2 x +1+i \sqrt {3}}\right )^{\frac {1}{4}} \left (\int \frac {\sqrt {\frac {3+i \left (2 x +1\right ) \sqrt {3}}{\sqrt {x^{2}+x +1}}}}{\sqrt {x -1}\, \left (x^{2}+x +1\right )^{\frac {3}{4}} \sqrt {\frac {\left (\sqrt {3}-3 i\right ) \sqrt {\left (x -1\right ) \left (\sqrt {3}+i\right ) \left (\sqrt {3}+2 i x +i\right )}-3 i \sqrt {3}+6 x -3}{2 x +1+i \sqrt {3}}}}d x \right )}{6 {\left (\frac {3+i \left (2 x +1\right ) \sqrt {3}}{\sqrt {x^{2}+x +1}}\right )}^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (x^{3}-1\right )^{\frac {1}{4}} \left (c_{1} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )\right ) \]

Problem 9734

ODE	\[ \boxed {y^{\prime \prime }+\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }+\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )}=0} \]
program solution
Maple solution	\[ \text {Expression too large to display} \]

Problem 9735

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (x^{2} \left (\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right )+\left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )+\left (x^{2}-\operatorname {a3} \right ) \left (x^{2}-\operatorname {a1} \right )\right )-\left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )\right ) y^{\prime }}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}+\frac {\left (A \,x^{2}+B \right ) y}{x \left (x^{2}-\operatorname {a1} \right ) \left (x^{2}-\operatorname {a2} \right ) \left (x^{2}-\operatorname {a3} \right )}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9736

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

Problem 9737

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

Problem 9738

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

Problem 9739

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

Problem 9740

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

Problem 9741

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

Problem 9742

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 n \cosh \left (x \right ) y^{\prime }}{\sinh \left (x \right )}+\left (-a^{2}+n^{2}\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \sinh \left (x \right )^{-n +\frac {1}{2}} \left (c_{1} \operatorname {LegendreP}\left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (a -\frac {1}{2}, n -\frac {1}{2}, \cosh \left (x \right )\right )\right ) \]

Problem 9743

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (1+2 n \right ) \cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\left (v +n +1\right ) \left (v -n \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \sin \left (x \right )^{-n} \left (c_{1} \operatorname {LegendreP}\left (v , n , \cos \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (v , n , \cos \left (x \right )\right )\right ) \]

Problem 9744

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

Problem 9745

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

Problem 9746

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

Problem 9747

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

Problem 9748

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

Problem 9749

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

Problem 9750

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

Problem 9751

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

Problem 9752

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

Problem 9753

ODE	\[ \boxed {y^{\prime \prime } \sin \left (x \right )^{2}-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (-3+2 a \right )^{2}}+3 a +2\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {-6+4 a +\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}}{-12+8 a}} \left (\cos \left (\frac {x}{2}\right )^{\frac {-6+4 a +\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}} \operatorname {hypergeom}\left (\left [\frac {8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-8 a -6}{-12+8 a}, \frac {-8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+16 a -6}{-12+8 a}\right ], \left [\frac {-6+4 a +\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) c_{2} +\cos \left (\frac {x}{2}\right )^{\frac {-6+4 a -\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}} \operatorname {hypergeom}\left (\left [\frac {8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-8 a -6}{-12+8 a}, \frac {-8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+16 a -6}{-12+8 a}\right ], \left [\frac {-6+4 a -\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{-6+4 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) c_{1} \right )}{\sqrt {\sin \left (x \right )}} \]

Problem 9754

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (-\left (b^{2} a^{2}-\left (1+a \right )^{2}\right ) \sin \left (x \right )^{2}-a \left (1+a \right ) b \sin \left (2 x \right )-a \left (-1+a \right )\right ) y}{\sin \left (x \right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (c_{2} \left (\cot \left (x \right )+i\right )^{\frac {1}{2}+\frac {1}{2} a +\frac {1}{2} i a b} \operatorname {hypergeom}\left (\left [i a b -a +1, a \left (i b +1\right )\right ], \left [i a b +a +2\right ], \frac {1}{2}-\frac {i \cot \left (x \right )}{2}\right )+c_{1} \left (\cot \left (x \right )+i\right )^{-\frac {1}{2}-\frac {1}{2} i a b -\frac {1}{2} a} \left (b +\cot \left (x \right )\right )\right ) \left (\cot \left (x \right )-i\right )^{-\frac {1}{2}+\frac {1}{2} i a b -\frac {1}{2} a} \]

Problem 9755

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

Problem 9756

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

Problem 9757

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

Problem 9758

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

Problem 9759

ODE	\[ \boxed {y^{\prime \prime }+\frac {\cos \left (x \right ) y^{\prime }}{\sin \left (x \right )}+\frac {\left (-17 \sin \left (x \right )^{2}-1\right ) y}{4 \sin \left (x \right )^{2}}=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{2 x}+c_{2} {\mathrm e}^{-2 x}}{\sqrt {\sin \left (x \right )}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sinh \left (2 x \right )+c_{2} \cosh \left (2 x \right )}{\sqrt {\sin \left (x \right )}} \]

Problem 9760

ODE	\[ \boxed {y^{\prime \prime }+\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}+\frac {\left (2 x^{2}+\sin \left (x \right )^{2} x^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}=\sqrt {\cos \left (x \right )}} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {\cos \left (x \right )}\, \left (4 c_{1} x^{5}-x^{4}+4 c_{2} \right )}{4 x^{2}} \]

Problem 9761

ODE	\[ \boxed {y^{\prime \prime }+\frac {b \cos \left (x \right ) y^{\prime }}{\sin \left (x \right ) a}+\frac {\left (c \cos \left (x \right )^{2}+d \cos \left (x \right )+e \right ) y}{a \sin \left (x \right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {2}\, \sin \left (x \right )^{-\frac {a +b}{2 a}} \sqrt {\cos \left (x \right )-1}\, \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}} \left (c_{1} \cos \left (\frac {x}{2}\right )^{-\frac {-2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-2 i \sqrt {4 a c -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+2 i \sqrt {4 a c -b^{2}}-\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [1-\frac {\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )+c_{2} \cos \left (\frac {x}{2}\right )^{\frac {2 a +\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}-2 i \sqrt {4 a c -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {a^{2}+\left (-2 b -4 c -4 d -4 e \right ) a +b^{2}}+2 i \sqrt {4 a c -b^{2}}+\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [1+\frac {\sqrt {a^{2}+\left (-2 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right )\right )}{2} \]

Problem 9762

ODE	\[ \boxed {y^{\prime \prime }+\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \sqrt {\sin \left (x \right )}\, \left (c_{1} \operatorname {LegendreP}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )\right ) \]

Problem 9763

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (4 v \left (v +1\right ) \sin \left (x \right )^{2}-\cos \left (x \right )^{2}+2-4 n^{2}\right ) y}{4 \sin \left (x \right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {\cos \left (x \right )}\, \left (-\frac {1}{2}+\frac {\cos \left (2 x \right )}{2}\right )^{\frac {n}{2}+\frac {1}{2}} \left (c_{1} \operatorname {hypergeom}\left (\left [-\frac {v}{2}+\frac {n}{2}, \frac {1}{2}+\frac {v}{2}+\frac {n}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+c_{2} \cos \left (x \right ) \operatorname {hypergeom}\left (\left [1+\frac {v}{2}+\frac {n}{2}, \frac {1}{2}-\frac {v}{2}+\frac {n}{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )\right )}{\sqrt {\sin \left (2 x \right )}} \]

Problem 9764

ODE	\[ \boxed {y^{\prime \prime }-\frac {\left (3 \sin \left (x \right )^{2}+1\right ) y^{\prime }}{\cos \left (x \right ) \sin \left (x \right )}-\frac {y \sin \left (x \right )^{2}}{\cos \left (x \right )^{2}}=0} \]
program solution	\[ y = c_{1} 3^{\frac {\sqrt {13}}{6}-\frac {1}{2}} \left (\sec \left (x \right )^{3}\right )^{\frac {1}{2}-\frac {\sqrt {13}}{6}}+c_{2} 3^{-\frac {1}{2}-\frac {\sqrt {13}}{6}} \left (\sec \left (x \right )^{3}\right )^{\frac {1}{2}+\frac {\sqrt {13}}{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \cos \left (x \right )^{-\frac {3}{2}+\frac {\sqrt {13}}{2}}+c_{2} \cos \left (x \right )^{-\frac {3}{2}-\frac {\sqrt {13}}{2}} \]

Problem 9765

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (-a \cos \left (x \right )^{2} \sin \left (x \right )^{2}-m \sin \left (x \right )^{2} \left (m -1\right )-n \left (-1+n \right ) \cos \left (x \right )^{2}\right ) y}{\cos \left (x \right )^{2} \sin \left (x \right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \sin \left (x \right )^{n} \left (c_{1} \cos \left (x \right )^{m} \operatorname {hypergeom}\left (\left [\frac {n}{2}+\frac {m}{2}+\frac {i \sqrt {a}}{2}, \frac {n}{2}+\frac {m}{2}-\frac {i \sqrt {a}}{2}\right ], \left [\frac {1}{2}+m \right ], \cos \left (x \right )^{2}\right )+c_{2} \cos \left (x \right )^{-m +1} \operatorname {hypergeom}\left (\left [\frac {n}{2}-\frac {m}{2}+\frac {i \sqrt {a}}{2}+\frac {1}{2}, \frac {n}{2}-\frac {m}{2}-\frac {i \sqrt {a}}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}-m \right ], \cos \left (x \right )^{2}\right )\right ) \]

Problem 9766

ODE	\[ \boxed {y^{\prime \prime }-\frac {\phi ^{\prime }\left (x \right ) y^{\prime }}{\phi \left (x \right )-\phi \left (a \right )}+\frac {\left (-n \left (1+n \right ) \left (\phi \left (x \right )-\phi \left (a \right )\right )^{2}+\frac {d^{2}}{d a^{2}}\phi \left (a \right )\right ) y}{\phi \left (x \right )-\phi \left (a \right )}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9767

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

Problem 9768

ODE	\[ \boxed {y^{\prime \prime }-\frac {2 \,\operatorname {JacobiSN}\left (x , k\right ) \operatorname {JacobiCN}\left (x , k\right ) \operatorname {JacobiDN}\left (x , k\right ) y^{\prime }-2 \left (1-2 \left (k^{2}+1\right ) \operatorname {JacobiSN}\left (a , k\right )^{2}+3 k^{2} \operatorname {JacobiSN}\left (a , k\right )^{4}\right ) y}{\operatorname {JacobiSN}\left (x , k\right )^{2}-\operatorname {JacobiSN}\left (a , k\right )}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9769

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

Problem 9770

ODE	\[ \boxed {y^{\prime \prime }+\frac {f^{\prime }\left (x \right ) y^{\prime }}{2 f \left (x \right )}+\frac {g \left (x \right ) y}{f \left (x \right )}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9771

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}+\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 g \left (x \right ) f^{\prime }\left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = f \left (x \right ) \left (\operatorname {LegendreQ}\left (v , g \left (x \right )\right ) c_{2} +\operatorname {LegendreP}\left (v , g \left (x \right )\right ) c_{1} \right ) \]

Problem 9772

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

Problem 9773

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

Problem 9774

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

Problem 9775

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

Problem 9776

ODE	\[ \boxed {y^{\prime \prime \prime }+y a \,x^{3}=b x} \]
program solution
Maple solution	\[ \text {Expression too large to display} \]

Problem 9777

ODE	\[ \boxed {y^{\prime \prime \prime }-a \,x^{b} y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +1}{b +3}, \frac {b +2}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +2}{b +3}, \frac {4+b}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {b +5}{b +3}, \frac {4+b}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right ) \]

Problem 9778

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

Problem 9779

ODE	\[ \boxed {y^{\prime \prime \prime }-a^{2} y^{\prime }={\mathrm e}^{2 x a} \sin \left (x \right )^{2}} \]
program solution	\[ y = c_{1} +{\mathrm e}^{x a} c_{2} +{\mathrm e}^{-x a} c_{3} +\frac {{\mathrm e}^{2 x a}}{12 a^{3}}+\frac {\left (-3 a^{3}+12 a \right ) {\mathrm e}^{2 x a} \cos \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64}+\frac {\left (-11 a^{2}+4\right ) {\mathrm e}^{2 x a} \sin \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (-9 a^{6}+36 a^{4}\right ) \cos \left (2 x \right )+\left (-33 a^{5}+12 a^{3}\right ) \sin \left (2 x \right )+9 a^{6}+49 a^{4}+56 a^{2}+16\right ) {\mathrm e}^{2 a x}+108 \left (a^{2}+\frac {4}{9}\right ) a^{2} \left (a^{2}+1\right ) \left (c_{3} a +c_{1} {\mathrm e}^{a x}-c_{2} {\mathrm e}^{-a x}\right ) \left (a^{2}+4\right )}{108 \left (a^{2}+\frac {4}{9}\right ) a^{3} \left (a^{2}+1\right ) \left (a^{2}+4\right )} \]

Problem 9780

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

Problem 9781

ODE	\[ \boxed {y^{\prime \prime \prime }-x^{2} y^{\prime \prime }+\left (a +b -1\right ) x y^{\prime }-b y a=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {a}{3}, -\frac {b}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], \frac {x^{3}}{3}\right )+c_{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}-\frac {a}{3}, \frac {1}{3}-\frac {b}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{3}\right ) x +c_{3} \operatorname {hypergeom}\left (\left [-\frac {a}{3}+\frac {2}{3}, -\frac {b}{3}+\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{3}\right ) x^{2} \]

Problem 9782

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

Problem 9783

ODE	\[ \boxed {y^{\prime \prime \prime }-3 \left (2 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+y b=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9784

ODE	\[ \boxed {y^{\prime \prime \prime }+\left (-n^{2}+1\right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\frac {\left (\left (-n^{2}+1\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right )-a \right ) y}{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9785

ODE	\[ \boxed {y^{\prime \prime \prime }-\left (4 n \left (1+n \right ) \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }-2 n \left (1+n \right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9786

ODE	\[ \boxed {y^{\prime \prime \prime }+\left (A \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )+a \right ) y^{\prime }+B \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9787

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

Problem 9788

ODE	\[ \boxed {y^{\prime \prime \prime }-\left (6 k^{2} \sin \left (x \right )^{2}+a \right ) y^{\prime }+y b=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9789

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

Problem 9790

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

Problem 9791

ODE	\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }-a^{2} y^{\prime }+2 a^{2} y=\sinh \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+{\mathrm e}^{x a} c_{2} +{\mathrm e}^{-x a} c_{3} +\frac {-3+4 \left (-a^{2} {\mathrm e}^{2 x}+{\mathrm e}^{2 x}-9\right ) \cosh \left (\frac {x}{2}\right )^{4}+4 \sinh \left (\frac {x}{2}\right ) \left (a^{2} {\mathrm e}^{2 x}-{\mathrm e}^{2 x}-9\right ) \cosh \left (\frac {x}{2}\right )^{3}+6 \left (a^{2} {\mathrm e}^{2 x}-{\mathrm e}^{2 x}+6\right ) \cosh \left (\frac {x}{2}\right )^{2}+18 \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right )}{3 \left (a^{2}-4\right ) \left (a^{2}-1\right ) \left (2 \cosh \left (\frac {x}{2}\right )^{2}+2 \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right )-1\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 c_{3} \left (a^{4}-5 a^{2}+4\right ) {\mathrm e}^{-a x}+2 \left (a +1\right ) \left (a^{2} c_{1} +\frac {\sinh \left (3 x \right )}{6}-4 c_{1} -\frac {\cosh \left (3 x \right )}{6}\right ) \left (a -1\right ) {\mathrm e}^{2 x}+2 c_{2} \left (a^{4}-5 a^{2}+4\right ) {\mathrm e}^{a x}+a^{2} {\mathrm e}^{x}-4 \,{\mathrm e}^{x}+{\mathrm e}^{-x}}{2 a^{4}-10 a^{2}+8} \]

Problem 9792

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

Problem 9793

ODE	\[ \boxed {y^{\prime \prime \prime }+\operatorname {a2} y^{\prime \prime }+\operatorname {a1} y^{\prime }+\operatorname {a0} y=0} \]
program solution	\[ \text {Expression too large to display} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \left (\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {2}{3}}+\frac {\operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{3}+\left (\operatorname {a1} -\frac {\operatorname {a2}^{2}}{3}\right ) \left (i \sqrt {3}-1\right )\right )}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a2}^{2}+12 i \sqrt {3}\, \operatorname {a1} -\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {2}{3}}-4 \operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}-4 \operatorname {a2}^{2}+12 \operatorname {a1} \right ) x}{12 \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}}+c_{3} {\mathrm e}^{\frac {\left (\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {2}{3}}-2 \operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a2}^{2}-12 \operatorname {a1} \right ) x}{6 \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}} \]

Problem 9794

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime } x +2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 y a x=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x^{2} \left (\operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2} c_{2} +\operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) \operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) c_{3} +\operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2} c_{1} \right ) \]

Problem 9795

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

Problem 9796

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

Problem 9797

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

Problem 9798

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

Problem 9799

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

Problem 9800

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime } f \left (x \right )+\left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}+4 g \left (x \right )\right ) y^{\prime }+\left (4 g \left (x \right ) f \left (x \right )+2 g^{\prime }\left (x \right )\right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]