Problem 9801
| ODE |
\[ \boxed {4 y^{\prime \prime \prime }-8 y^{\prime \prime }-11 y^{\prime }-3 y=-18 \,{\mathrm e}^{x}} \] |
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program solution |
\[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{-\frac {x}{2}}+c_{1} {\mathrm e}^{3 x}+{\mathrm e}^{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{-\frac {x}{2}}+c_{1} {\mathrm e}^{3 x}+{\mathrm e}^{x} \] |
Problem 9802
| ODE |
\[ \boxed {27 y^{\prime \prime \prime }-36 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime } n^{2}-2 n \left (n +3\right ) \left (4 n -3\right ) \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9803
| ODE |
\[ \boxed {y^{\prime \prime \prime } x +3 y^{\prime \prime }+x y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\left ({\mathrm e}^{\frac {3 x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{3} +{\mathrm e}^{\frac {3 x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_{2} +c_{1} \right ) {\mathrm e}^{-x}}{x} \] |
Problem 9804
| ODE |
\[ \boxed {y^{\prime \prime \prime } x +3 y^{\prime \prime }-a \,x^{2} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{4}, \frac {5}{4}\right ], \frac {x^{4} a}{64}\right )+\frac {c_{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {3}{4}\right ], \frac {x^{4} a}{64}\right )}{x}+c_{3} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {5}{4}, \frac {3}{2}\right ], \frac {x^{4} a}{64}\right ) \] |
Problem 9805
| ODE |
\[ \boxed {y^{\prime \prime \prime } x +\left (a +b \right ) y^{\prime \prime }-y^{\prime } x -a y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\frac {a}{2}\right ], \left [\frac {1}{2}, \frac {a}{2}+\frac {b}{2}\right ], \frac {x^{2}}{4}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {a}{2}\right ], \left [\frac {3}{2}, \frac {a}{2}+\frac {b}{2}+\frac {1}{2}\right ], \frac {x^{2}}{4}\right )+c_{3} x^{-a -b +2} \operatorname {hypergeom}\left (\left [1-\frac {b}{2}\right ], \left [2-\frac {b}{2}-\frac {a}{2}, -\frac {a}{2}-\frac {b}{2}+\frac {3}{2}\right ], \frac {x^{2}}{4}\right ) \] |
Problem 9806
| ODE |
\[ \boxed {y^{\prime \prime \prime } x -\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (x -1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {x^{2+v} c_{2} \operatorname {BesselI}\left (-v +1, x\right )-2 \operatorname {BesselI}\left (-v , x\right ) x^{v +1} c_{2} v +x^{2+v} c_{3} \operatorname {BesselK}\left (v +1, x\right )+{\mathrm e}^{x} c_{1} x}{x} \] |
Problem 9807
| ODE |
\[ \boxed {y^{\prime \prime \prime } x +\left (x^{2}-3\right ) y^{\prime \prime }+4 y^{\prime } x +2 y=f \left (x \right )} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (c_{3} +\int \frac {\left (2 c_{1} x +c_{2} +\int \int f \left (x \right )d x d x \right ) {\mathrm e}^{\frac {x^{2}}{2}}}{x^{6}}d x \right ) {\mathrm e}^{-\frac {x^{2}}{2}} x^{5} \] |
Problem 9808
| ODE |
\[ \boxed {2 y^{\prime \prime \prime } x +3 y^{\prime \prime }+y a x=b} \] |
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program solution |
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 9809
| ODE |
\[ \boxed {2 y^{\prime \prime \prime } x -4 \left (x +\nu -1\right ) y^{\prime \prime }+\left (2 x +6 \nu -5\right ) y^{\prime }+\left (1-2 \nu \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{\frac {x}{2}} x^{\nu } \operatorname {BesselI}\left (\nu , \frac {x}{2}\right )+c_{3} {\mathrm e}^{\frac {x}{2}} x^{\nu } \operatorname {BesselK}\left (\nu , \frac {x}{2}\right ) \] |
Problem 9810
| ODE |
\[ \boxed {2 y^{\prime \prime \prime } x +3 \left (2 x a +k \right ) y^{\prime \prime }+6 \left (a k +b x \right ) y^{\prime }+\left (3 b k +2 c x \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9811
| ODE |
\[ \boxed {\left (x -2\right ) x y^{\prime \prime \prime }-\left (x -2\right ) x y^{\prime \prime }-2 y^{\prime }+2 y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{3} \operatorname {expIntegral}_{1}\left (x -2\right ) {\mathrm e}^{x -2}+\frac {c_{3} x^{2} \ln \left (x -2\right )}{4}+{\mathrm e}^{x} c_{2} -\frac {c_{3} x^{2} \ln \left (x \right )}{4}+\frac {\left (2 x +2\right ) c_{3}}{4}+c_{1} x^{2} \] |
Problem 9812
| ODE |
\[ \boxed {\left (2 x -1\right ) y^{\prime \prime \prime }-8 y^{\prime } x +8 y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} x +c_{2} {\mathrm e}^{2 x}-\frac {c_{3} x \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-1+2 x \right )}{2}+\frac {c_{3} \operatorname {expIntegral}_{1}\left (-2+4 x \right ) {\mathrm e}^{2 x -2}}{4}+\frac {c_{3} {\mathrm e}^{-2 x}}{4} \] |
Problem 9813
| ODE |
\[ \boxed {\left (2 x -1\right ) y^{\prime \prime \prime }+\left (4+x \right ) y^{\prime \prime }+2 y^{\prime }=0} \] |
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program solution |
\[ y = \int \frac {x 2^{\frac {1}{8}} {\mathrm e}^{-\frac {x}{2}} \left (c_{2} 2^{\frac {3}{4}} \left (\int \frac {{\mathrm e}^{\frac {x}{2}} \left (2 x -1\right )^{\frac {1}{4}}}{x^{2}}d x \right )+2 c_{1} \right )}{2 \left (2 x -1\right )^{\frac {5}{4}}}d x +c_{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (c_{3} +\int \frac {\left (2 c_{1} x +c_{2} \right ) {\mathrm e}^{\frac {x}{2}}}{\left (-1+2 x \right )^{\frac {3}{4}}}d x \right ) {\mathrm e}^{-\frac {x}{2}}}{\left (-1+2 x \right )^{\frac {1}{4}}} \] |
Problem 9814
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-6 y^{\prime }+a \,x^{2} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {-c_{3} \left (\left (-i-\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {2}{3}}+i x \,a^{3}\right ) {\mathrm e}^{-\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) x}{2 a}}+\left (\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {2}{3}}+i x \,a^{3}\right ) c_{2} {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) x}{2 a}}+c_{1} \left (x \,a^{3}+2 \left (-a^{4}\right )^{\frac {2}{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} x}{a}}}{x} \] |
Problem 9815
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}+\left (x +1\right ) y^{\prime \prime }-y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9816
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-y^{\prime \prime } x +\left (x^{2}+1\right ) y^{\prime }=0} \] |
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program solution |
\[ y = c_{1} x \operatorname {BesselJ}\left (1, x\right )+c_{2} x \operatorname {BesselY}\left (1, x\right )+c_{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +c_{2} x \operatorname {BesselJ}\left (1, x\right )+c_{3} x \operatorname {BesselY}\left (1, x\right ) \] |
Problem 9817
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}+3 y^{\prime \prime } x +\left (4 a^{2} x^{2 a}+1-4 \nu ^{2} a^{2}\right ) y^{\prime }-4 a^{3} x^{2 a -1} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}\right ], \left [\nu +1, -\nu +1\right ], -x^{2 a}\right )+c_{2} x^{-2 a \nu } \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\nu \right ], \left [-\nu +1, 1-2 \nu \right ], -x^{2 a}\right )+c_{3} x^{2 a \nu } \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\nu \right ], \left [\nu +1, 2 \nu +1\right ], -x^{2 a}\right ) \] |
Problem 9818
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-3 \left (x -m \right ) x y^{\prime \prime }+\left (2 x^{2}+4 \left (n -m \right ) x +m \left (-1+2 m \right )\right ) y^{\prime }-2 n \left (2 x -2 m +1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {KummerM}\left (-n , m , x\right )^{2}+c_{2} \operatorname {KummerU}\left (-n , m , x\right )^{2}+c_{3} \operatorname {KummerM}\left (-n , m , x\right ) \operatorname {KummerU}\left (-n , m , x\right ) \] |
Problem 9819
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}+4 y^{\prime \prime } x +\left (x^{2}+2\right ) y^{\prime }+3 x y=f \left (x \right )} \] |
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program solution |
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 9820
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}+5 y^{\prime \prime } x +4 y^{\prime }=\ln \left (x \right )} \] |
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program solution |
\[ y = \frac {-c_{2} \ln \left (x \right )-c_{2} -c_{1}}{x}+c_{3} +\frac {x \left (-2+\ln \left (x \right )\right )}{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (x^{2}+4 c_{2} \right ) \ln \left (x \right )-2 x^{2}+4 c_{1} x +4 c_{3}}{4 x} \] |
Problem 9821
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}+6 y^{\prime \prime } x +6 y^{\prime }=0} \] |
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program solution |
\[ y = -\frac {2 c_{2} x +c_{1}}{2 x^{2}}+c_{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +\frac {c_{2}}{x^{2}}+\frac {c_{3}}{x} \] |
Problem 9822
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}+6 y^{\prime \prime } x +6 y^{\prime }+a \,x^{2} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\frac {\left (-a \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) x}{2}}+c_{2} {\mathrm e}^{-\frac {\left (-a \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) x}{2}}+c_{3} {\mathrm e}^{\left (-a \right )^{\frac {1}{3}} x}}{x^{2}} \] |
Problem 9823
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-3 \left (p +q \right ) x y^{\prime \prime }+3 p \left (3 q +1\right ) y^{\prime }-y x^{2}=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [-p +\frac {2}{3}, -q +\frac {1}{3}\right ], \frac {x^{3}}{27}\right )+c_{2} x^{2+3 q} \operatorname {hypergeom}\left (\left [\right ], \left [q +\frac {5}{3}, \frac {4}{3}+q -p \right ], \frac {x^{3}}{27}\right )+c_{3} x^{3 p +1} \operatorname {hypergeom}\left (\left [\right ], \left [p +\frac {4}{3}, \frac {2}{3}-q +p \right ], \frac {x^{3}}{27}\right ) \] |
Problem 9824
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-2 \left (1+n \right ) x y^{\prime \prime }+\left (x^{2} a +6 n \right ) y^{\prime }-2 y a x=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} x^{n +\frac {1}{2}} \operatorname {BesselJ}\left (-n -\frac {1}{2}, x \sqrt {a}\right )+c_{2} x^{n +\frac {1}{2}} \operatorname {BesselY}\left (-n -\frac {1}{2}, x \sqrt {a}\right )+c_{3} \left (a \,x^{2}+4 n -2\right ) \] |
Problem 9825
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-\left (x^{2}-2 x \right ) y^{\prime \prime }-\left (x^{2}+\nu ^{2}-\frac {1}{4}\right ) y^{\prime }+\left (x^{2}-2 x +\nu ^{2}-\frac {1}{4}\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x}+c_{2} \sqrt {x}\, \operatorname {BesselI}\left (\nu , x\right )+c_{3} \sqrt {x}\, \operatorname {BesselK}\left (\nu , x\right ) \] |
Problem 9826
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-\left (x +\nu \right ) x y^{\prime \prime }+\nu \left (2 x +1\right ) y^{\prime }-\nu \left (x +1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {-c_{3} x^{\frac {\nu }{2}+1} \operatorname {BesselY}\left (-\nu +1, 2 \sqrt {\nu }\, \sqrt {x}\right )-c_{2} x^{\frac {\nu }{2}+1} \operatorname {BesselJ}\left (-\nu +1, 2 \sqrt {\nu }\, \sqrt {x}\right )-x^{\frac {\nu }{2}+\frac {1}{2}} \sqrt {\nu }\, \operatorname {BesselY}\left (-\nu , 2 \sqrt {\nu }\, \sqrt {x}\right ) c_{3} -x^{\frac {\nu }{2}+\frac {1}{2}} \sqrt {\nu }\, \operatorname {BesselJ}\left (-\nu , 2 \sqrt {\nu }\, \sqrt {x}\right ) c_{2} +c_{1} {\mathrm e}^{x} \sqrt {x}}{\sqrt {x}} \] |
Problem 9827
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-2 \left (x^{2}-x \right ) y^{\prime \prime }+\left (x^{2}-2 x +\frac {1}{4}-\nu ^{2}\right ) y^{\prime }+\left (\nu ^{2}-\frac {1}{4}\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{\frac {x}{2}} \sqrt {x}\, \operatorname {BesselI}\left (\nu , \frac {x}{2}\right )+c_{3} {\mathrm e}^{\frac {x}{2}} \sqrt {x}\, \operatorname {BesselK}\left (\nu , \frac {x}{2}\right ) \] |
Problem 9828
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{2}-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 y x^{2}=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{3} \left (\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselK}\left (\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+2 \operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x \right )+c_{2} \left (\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x \right )+c_{1}}{x^{2}} \] |
Problem 9829
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime \prime }+8 y^{\prime \prime } x +10 y^{\prime }=3-\frac {1}{x^{2}}+2 \ln \left (x \right )} \] |
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program solution |
\[ y = \frac {-\frac {1}{3} c_{2} x^{3}-c_{2} x -\frac {1}{4} c_{1}}{\left (x^{2}+1\right )^{2}}+c_{3} +\frac {\left (45 x^{5}+150 x^{3}+225 x \right ) \ln \left (x \right )-9 x^{5}+25 x^{3}}{225 \left (x^{2}+1\right )^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (45 x^{5}+150 x^{3}+225 x \right ) \ln \left (x \right )-9 x^{5}+225 c_{1} x^{4}+\left (225 c_{2} -50\right ) x^{3}+450 c_{1} x^{2}+\left (675 c_{2} -225\right ) x +225 c_{3}}{225 \left (x^{2}+1\right )^{2}} \] |
Problem 9830
| ODE |
\[ \boxed {\left (x^{2}+2\right ) y^{\prime \prime \prime }-2 y^{\prime \prime } x +\left (x^{2}+2\right ) y^{\prime }-2 x y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} x^{2}+c_{2} \cos \left (x \right )+c_{3} \sin \left (x \right ) \] |
Problem 9831
| ODE |
\[ \boxed {2 x \left (x -1\right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 x a +b \right ) y^{\prime }+a y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {MathieuC}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}+c_{2} \operatorname {MathieuS}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}+c_{3} \operatorname {MathieuC}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \operatorname {MathieuS}\left (1-\frac {a}{2}-\frac {b}{2}, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \] |
Problem 9832
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+\left (-\nu ^{2}+1\right ) x y^{\prime }+\left (a \,x^{3}+\nu ^{2}-1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x \left (c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {\nu }{3}, 1+\frac {\nu }{3}\right ], -\frac {x^{3} a}{27}\right )+c_{2} x^{-\nu } \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {2 \nu }{3}, 1-\frac {\nu }{3}\right ], -\frac {x^{3} a}{27}\right )+c_{3} x^{\nu } \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {2 \nu }{3}, 1+\frac {\nu }{3}\right ], -\frac {x^{3} a}{27}\right )\right ) \] |
Problem 9833
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+\left (4 x^{3}+\left (-4 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (4 \nu ^{2}-1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x \left (c_{1} \operatorname {BesselJ}\left (\nu , x\right )^{2}+c_{2} \operatorname {BesselY}\left (\nu , x\right )^{2}+c_{3} \operatorname {BesselJ}\left (\nu , x\right ) \operatorname {BesselY}\left (\nu , x\right )\right ) \] |
Problem 9834
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+\left (a \,x^{2 \nu }+1-\nu ^{2}\right ) x y^{\prime }+\left (b \,x^{3 \nu }+a \left (\nu -1\right ) x^{2 \nu }+\nu ^{2}-1\right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9835
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+3 x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y=6 x^{3} \left (x -1\right ) \ln \left (x \right )-x^{3} \left (x +8\right )} \] |
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program solution |
\[ y = \frac {c_{1}}{x^{2}}+c_{2} x +c_{3} \ln \left (x \right ) x +\frac {x^{3} \left (50 x \ln \left (x \right )-135 \ln \left (x \right )-50 x -18\right )}{450} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (50 x^{6}-135 x^{5}+450 c_{3} x^{3}\right ) \ln \left (x \right )-50 x^{6}-18 x^{5}+450 c_{1} x^{3}+450 c_{2}}{450 x^{2}} \] |
Problem 9836
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+3 x^{2} y^{\prime \prime }+\left (-a^{2}+1\right ) x y^{\prime }=0} \] |
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program solution |
\[ y = \frac {c_{1} x^{a}}{a}-\frac {c_{2} x^{-a}}{a}+c_{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +c_{2} x^{a}+c_{3} x^{-a} \] |
Problem 9837
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}-4 x^{2} y^{\prime \prime }+\left (x^{2}+8\right ) x y^{\prime }-2 \left (x^{2}+4\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x \left (\cos \left (x \right ) c_{3} +\sin \left (x \right ) c_{2} +c_{1} x \right ) \] |
Problem 9838
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+6 x^{2} y^{\prime \prime }+\left (a \,x^{3}-12\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {-c_{2} \left (\left (-i-\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {2}{3}}+i x \,a^{3}\right ) {\mathrm e}^{-\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) x}{2 a}}+c_{3} \left (\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {2}{3}}+i x \,a^{3}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) x}{2 a}}+c_{1} \left (x \,a^{3}+2 \left (-a^{4}\right )^{\frac {2}{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} x}{a}}}{x^{3}} \] |
Problem 9839
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+3 \left (1-a \right ) x^{2} y^{\prime \prime }+\left (4 b^{2} c^{2} x^{1+2 c}+1-4 \nu ^{2} c^{2}+3 a \left (-1+a \right ) x \right ) y^{\prime }+\left (4 b^{2} c^{2} \left (-a +c \right ) x^{2 c}+a \left (4 \nu ^{2} c^{2}-a^{2}\right )\right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9840
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+\left (x +3\right ) x^{2} y^{\prime \prime }+5 \left (x -6\right ) x y^{\prime }+\left (4 x +30\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{3} {\mathrm e}^{-x} \left (x^{8}+28 x^{7}+450 x^{6}+5100 x^{5}+42900 x^{4}+267120 x^{3}+1179360 x^{2}+3326400 x +4536000\right ) \operatorname {expIntegral}_{1}\left (-x \right )+c_{2} {\mathrm e}^{-x} \left (x^{8}+28 x^{7}+450 x^{6}+5100 x^{5}+42900 x^{4}+267120 x^{3}+1179360 x^{2}+3326400 x +4536000\right )+60 c_{3} \left (x^{4}-84 x^{3}+2016 x^{2}-20160 x +75600\right ) \ln \left (x \right )+c_{3} x^{7}+29 c_{3} x^{6}+480 x^{5} c_{3} +\left (c_{1} +5612 c_{3} \right ) x^{4}+\left (-84 c_{1} +40152 c_{3} \right ) x^{3}+\left (2016 c_{1} +654192 c_{3} \right ) x^{2}+\left (-20160 c_{1} -2761920 c_{3} \right ) x +75600 c_{1} +27367200 c_{3}}{x^{6}} \] |
Problem 9841
| ODE |
\[ \boxed {y^{\prime \prime \prime } x^{3}+x^{2} y^{\prime \prime }+2 y^{\prime } x -y=2 x^{3}-\ln \left (x \right )} \] |
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program solution |
\[ \text {Expression too large to display} \] Warning, solution could not be verified |
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 9842
| ODE |
\[ \boxed {\left (x^{2}+1\right ) x y^{\prime \prime \prime }+3 \left (2 x^{2}+1\right ) y^{\prime \prime }-12 y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right ) \sqrt {x^{2}+1}\, c_{2} x^{2}+c_{1} \sqrt {x^{2}+1}\, x^{2}+2 c_{3} x^{3}-3 c_{2} x^{2}+c_{3} x -c_{2}}{x} \] |
Problem 9843
| ODE |
\[ \boxed {\left (x +3\right ) x^{2} y^{\prime \prime \prime }-3 x \left (x +2\right ) y^{\prime \prime }+6 \left (x +1\right ) y^{\prime }-6 y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{2} x^{3}+c_{1} x^{2}+c_{3} x +c_{3} \] |
Problem 9844
| ODE |
\[ \boxed {2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a2} +\operatorname {a3} +\operatorname {a1} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (1+n \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -c_{2} \left (x -\operatorname {a1} \right ) {\operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +3\right ) \operatorname {a1} -b}{-4 \operatorname {a2} +4 \operatorname {a1}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right )}^{2}+c_{3} \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +1\right ) \operatorname {a1} -b +\operatorname {a2} +\operatorname {a3}}{-4 \operatorname {a2} +4 \operatorname {a1}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right ) \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +3\right ) \operatorname {a1} -b}{-4 \operatorname {a2} +4 \operatorname {a1}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right ) \sqrt {-x +\operatorname {a1}}+c_{1} {\operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{-\operatorname {a2} +\operatorname {a1}}, \frac {\left (-n^{2}-n +1\right ) \operatorname {a1} -b +\operatorname {a2} +\operatorname {a3}}{-4 \operatorname {a2} +4 \operatorname {a1}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {-x +\operatorname {a1}}{-\operatorname {a2} +\operatorname {a1}}\right )}^{2} \] |
Problem 9845
| ODE |
\[ \boxed {\left (x +1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x +2\right ) x^{2} y^{\prime \prime }+\left (10 x +4\right ) x y^{\prime }-4 \left (3 x +1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x \left (\ln \left (x \right )^{2} c_{3} x +c_{2} x \ln \left (x \right )+x^{2} c_{3} +c_{1} x +c_{3} \right ) \] |
Problem 9846
| ODE |
\[ \boxed {4 y^{\prime \prime \prime } x^{4}-4 x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }=1} \] |
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program solution |
\[ y = \left (\frac {c_{1}}{2}-\frac {c_{2}}{4}\right ) x^{2}+\frac {\ln \left (x \right ) c_{2} x^{2}}{2}+c_{3} -\frac {1}{36 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {18 \ln \left (x \right ) c_{1} x^{3}-1+\left (-9 c_{1} +18 c_{2} \right ) x^{3}+36 c_{3} x}{36 x} \] |
Problem 9847
| ODE |
\[ \boxed {\left (x^{2}+1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-4 \left (3 x^{2}+1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x \left (c_{2} x \ln \left (x \right )+x^{2} c_{3} +\left (c_{1} +c_{2} \right ) x +c_{3} \right ) \] |
Problem 9848
| ODE |
\[ \boxed {x^{6} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x^{2} \left (c_{1} +\left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{\frac {11}{2}}}d x \right ) c_{2} +\left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\operatorname {BesselK}\left (\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{\frac {11}{2}}}d x \right ) c_{3} \right ) \] |
Problem 9849
| ODE |
\[ \boxed {x^{6} y^{\prime \prime \prime }+6 y^{\prime \prime } x^{5}+a y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {4 \left (-8 x^{3}+a \right )^{4} \left (c_{3} {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 a x}} \left (\frac {\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {1}{3}}}{4}+i a x \right )+\left (\frac {\left (\sqrt {3}+i\right ) \left (-a^{4}\right )^{\frac {1}{3}}}{4}-i a x \right ) c_{2} {\mathrm e}^{-\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2 a x}}+128 \,{\mathrm e}^{-\frac {\left (-a^{4}\right )^{\frac {1}{3}}}{a x}} c_{1} \left (a x +\frac {\left (-a^{4}\right )^{\frac {1}{3}}}{2}\right )\right )}{{\left (2 a x +\left (-a^{4}\right )^{\frac {1}{3}}\right )}^{4} {\left (i \left (-a^{4}\right )^{\frac {1}{3}} \sqrt {3}-4 a x +\left (-a^{4}\right )^{\frac {1}{3}}\right )}^{4} {\left (i \left (-a^{4}\right )^{\frac {1}{3}} \sqrt {3}+4 a x -\left (-a^{4}\right )^{\frac {1}{3}}\right )}^{4}} \] |
Problem 9850
| ODE |
\[ \boxed {x^{2} \left (x^{4}+2 x^{2}+2 x +1\right ) y^{\prime \prime \prime }-\left (2 x^{6}+3 x^{4}-6 x^{2}-6 x -1\right ) y^{\prime \prime }+\left (x^{6}-6 x^{3}-15 x^{2}-12 x -2\right ) y^{\prime }+\left (x^{4}+4 x^{3}+8 x^{2}+6 x +1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{2} {\mathrm e}^{\frac {1}{x}}+{\mathrm e}^{x} \left (c_{3} x +c_{1} \right ) \] |
Problem 9851
| ODE |
\[ \boxed {\left (x -a \right )^{3} \left (x -b \right )^{3} y^{\prime \prime \prime }-y c=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (c_{1} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =1\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =1\right )}{a -b}}+c_{2} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =2\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =2\right )}{a -b}}+c_{3} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =3\right )}{a -b}} \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 b -3 a \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =3\right )}{a -b}}\right ) \] |
Problem 9852
| ODE |
\[ \boxed {y^{\prime \prime \prime } \sin \left (x \right )+\left (2 \cos \left (x \right )+1\right ) y^{\prime \prime }-y^{\prime } \sin \left (x \right )=\cos \left (x \right )} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) c_{1} -\ln \left (\sin \left (x \right )\right ) c_{1} -\cot \left (x \right )^{2} x +\left (c_{1} x +c_{2} +1\right ) \cot \left (x \right )+\csc \left (x \right )^{2} x +\left (-c_{1} x -c_{2} -1\right ) \csc \left (x \right )-c_{3}}{-\csc \left (x \right )+\cot \left (x \right )} \] |
Problem 9853
| ODE |
\[ \boxed {\left (\sin \left (x \right )+x \right ) y^{\prime \prime \prime }+3 \left (\cos \left (x \right )+1\right ) y^{\prime \prime }-3 y^{\prime } \sin \left (x \right )-y \cos \left (x \right )=-\sin \left (x \right )} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{3} +c_{1} x^{2}+x c_{2} -\cos \left (x \right )}{\sin \left (x \right )+x} \] |
Problem 9854
| ODE |
\[ \boxed {y^{\prime \prime \prime } \sin \left (x \right )^{2}+3 y^{\prime \prime } \sin \left (x \right ) \cos \left (x \right )+\left (\cos \left (2 x \right )+4 \nu \left (1+\nu \right ) \sin \left (x \right )^{2}\right ) y^{\prime }+2 \nu \left (1+\nu \right ) y \sin \left (2 x \right )=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{2}+c_{2} \cos \left (x \right )^{2} \operatorname {hypergeom}\left (\left [\frac {\nu }{2}+1, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{2}+c_{3} \operatorname {hypergeom}\left (\left [-\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \cos \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {\nu }{2}+1, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \] |
Problem 9855
| ODE |
\[ \boxed {f^{\prime }\left (x \right ) y^{\prime \prime }+f \left (x \right ) y^{\prime \prime \prime }+g^{\prime }\left (x \right ) y^{\prime }+g \left (x \right ) y^{\prime \prime }+h^{\prime }\left (x \right ) y+h \left (x \right ) y^{\prime }+A \left (x \right ) \left (y^{\prime \prime } f \left (x \right )+g \left (x \right ) y^{\prime }+h \left (x \right ) y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9856
| ODE |
\[ \boxed {y^{\prime \prime \prime }+y^{\prime } x +y n=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\frac {n}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], -\frac {x^{3}}{9}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\frac {1}{3}+\frac {n}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], -\frac {x^{3}}{9}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}+\frac {n}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], -\frac {x^{3}}{9}\right ) \] |
Problem 9857
| ODE |
\[ \boxed {y^{\prime \prime \prime }-y^{\prime } x -y n=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\frac {n}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], \frac {x^{3}}{9}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\frac {1}{3}+\frac {n}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{9}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}+\frac {n}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{9}\right ) \] |
Problem 9858
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }=0} \] |
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program solution |
\[ y = c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{6} c_{1} x^{3}+\frac {1}{2} c_{2} x^{2}+c_{3} x +c_{4} \] |
Problem 9859
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+4 y=f} \] |
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program solution |
\[ y = {\mathrm e}^{\left (1+i\right ) x} c_{1} +{\mathrm e}^{\left (-1-i\right ) x} c_{2} +{\mathrm e}^{\left (-1+i\right ) x} c_{3} +{\mathrm e}^{\left (1-i\right ) x} c_{4} +\frac {f}{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {f}{4}+\cos \left (x \right ) c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{x} \sin \left (x \right )+c_{3} {\mathrm e}^{-x} \cos \left (x \right )+c_{4} \sin \left (x \right ) {\mathrm e}^{-x} \] |
Problem 9860
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+\lambda y=0} \] |
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program solution |
\[ y = {\mathrm e}^{i \left (-\lambda \right )^{\frac {1}{4}} x} c_{1} +{\mathrm e}^{-i \left (-\lambda \right )^{\frac {1}{4}} x} c_{2} +{\mathrm e}^{\left (-\lambda \right )^{\frac {1}{4}} x} c_{3} +{\mathrm e}^{-\left (-\lambda \right )^{\frac {1}{4}} x} c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-i \left (-\lambda \right )^{\frac {1}{4}} x}+c_{2} {\mathrm e}^{i \left (-\lambda \right )^{\frac {1}{4}} x}+c_{3} {\mathrm e}^{-\left (-\lambda \right )^{\frac {1}{4}} x}+c_{4} {\mathrm e}^{\left (-\lambda \right )^{\frac {1}{4}} x} \] |
Problem 9861
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-12 y^{\prime \prime }+12 y=16 \,{\mathrm e}^{x^{2}} x^{4}} \] |
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program solution |
\[ y = {\mathrm e}^{-\sqrt {6+2 \sqrt {6}}\, x} c_{1} +{\mathrm e}^{\sqrt {6+2 \sqrt {6}}\, x} c_{2} +{\mathrm e}^{\sqrt {6-2 \sqrt {6}}\, x} c_{3} +{\mathrm e}^{-\sqrt {6-2 \sqrt {6}}\, x} c_{4} +{\mathrm e}^{x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{x^{2}}+c_{1} {\mathrm e}^{\sqrt {6-2 \sqrt {6}}\, x}+c_{2} {\mathrm e}^{\sqrt {6+2 \sqrt {6}}\, x}+c_{3} {\mathrm e}^{-\sqrt {6-2 \sqrt {6}}\, x}+c_{4} {\mathrm e}^{-\sqrt {6+2 \sqrt {6}}\, x} \] |
Problem 9862
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+2 a^{2} y^{\prime \prime }+a^{4} y=\cosh \left (x a \right )} \] |
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program solution |
\[ y = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{-i x a}+\left (c_{4} x +c_{3} \right ) {\mathrm e}^{i x a}+\frac {-\sinh \left (x a \right ) x +2 \left (\int \cosh \left (x a \right ) \left (\cos \left (x a \right ) a x -\sin \left (x a \right )\right )d x \right ) \cos \left (x a \right )+2 \left (\int \cosh \left (x a \right ) \left (\sin \left (x a \right ) x a +\cos \left (x a \right )\right )d x \right ) \sin \left (x a \right )}{4 a^{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{-a x}+\left (8 c_{3} x +8 c_{1} \right ) a^{4} \cos \left (a x \right )+\left (8 c_{4} x +8 c_{2} \right ) a^{4} \sin \left (a x \right )+{\mathrm e}^{a x}}{8 a^{4}} \] |
Problem 9863
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+\left (\lambda +1\right ) a^{2} y^{\prime \prime }+\lambda \,a^{4} y=0} \] |
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program solution |
\[ y = {\mathrm e}^{-i x a} c_{1} +{\mathrm e}^{-\sqrt {-\lambda }\, a x} c_{2} +{\mathrm e}^{i x a} c_{3} +{\mathrm e}^{\sqrt {-\lambda }\, a x} c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sin \left (a x \right )+c_{2} \cos \left (a x \right )+c_{3} \sin \left (a \sqrt {\lambda }\, x \right )+c_{4} \cos \left (a \sqrt {\lambda }\, x \right ) \] |
Problem 9864
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+a \left (b x -1\right ) y^{\prime \prime }+a b y^{\prime }+\lambda y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9865
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+\left (x^{2} a +b \lambda +c \right ) y^{\prime \prime }+\left (x^{2} a +\beta \lambda +\gamma \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9866
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+a \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime \prime }+b \operatorname {WeierstrassPPrime}\left (x , \operatorname {g2} , \operatorname {g3}\right ) y^{\prime }+\left (c \left (6 \operatorname {WeierstrassP}\left (x , \operatorname {g2} , \operatorname {g3}\right )^{2}-\frac {\operatorname {g2}}{2}\right )+d \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9867
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-\left (12 k^{2} \operatorname {JacobiSN}\left (z , x\right )^{2}+a \right ) y^{\prime \prime }+b y^{\prime }+\left (\alpha \operatorname {JacobiSN}\left (z , x\right )^{2}+\beta \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9868
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }+4 y=32 \sin \left (2 x \right )-24 \cos \left (2 x \right )} \] |
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program solution |
\[ y = \left (\left (c_{4} x +c_{3} \right ) {\mathrm e}^{3 x}+c_{2} x +c_{1} \right ) {\mathrm e}^{-2 x}+\sin \left (2 x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-2 x} \left (\left (c_{3} x +c_{1} \right ) {\mathrm e}^{3 x}+c_{4} x +\sin \left (2 x \right ) {\mathrm e}^{2 x}+c_{2} \right ) \] |
Problem 9869
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }+4 a x y^{\prime \prime \prime }+6 a^{2} x^{2} y^{\prime \prime }+4 a^{3} x^{3} y^{\prime }+a^{4} x^{4} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (2 \sqrt {3+\sqrt {6}}\, \sqrt {a}+2 \sqrt {3-\sqrt {6}}\, \sqrt {a}+a x \right )}{2}} \left (c_{2} {\mathrm e}^{\sqrt {a}\, x \left (\sqrt {3+\sqrt {6}}+2 \sqrt {3-\sqrt {6}}\right )}+c_{4} {\mathrm e}^{\sqrt {a}\, x \left (2 \sqrt {3+\sqrt {6}}+\sqrt {3-\sqrt {6}}\right )}+c_{3} {\mathrm e}^{\sqrt {3-\sqrt {6}}\, \sqrt {a}\, x}+c_{1} {\mathrm e}^{\sqrt {3+\sqrt {6}}\, \sqrt {a}\, x}\right ) \] |
Problem 9870
| ODE |
\[ \boxed {4 y^{\prime \prime \prime \prime }-12 y^{\prime \prime \prime }+11 y^{\prime \prime }-3 y^{\prime }=4 \cos \left (x \right )} \] |
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program solution |
\[ y = c_{1} +c_{2} {\mathrm e}^{x}+{\mathrm e}^{\frac {3 x}{2}} c_{3} +{\mathrm e}^{\frac {x}{2}} c_{4} -\frac {14 \cos \left (x \right )}{65}+\frac {18 \sin \left (x \right )}{65} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{x}+2 c_{2} {\mathrm e}^{\frac {x}{2}}+\frac {2 c_{3} {\mathrm e}^{\frac {3 x}{2}}}{3}+\frac {18 \sin \left (x \right )}{65}-\frac {14 \cos \left (x \right )}{65}+c_{4} \] |
Problem 9871
| ODE |
\[ \boxed {x y^{\prime \prime \prime \prime }+5 y^{\prime \prime \prime }=24} \] |
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program solution |
\[ y = \frac {c_{1} x^{2}}{8}+c_{4} x +\frac {{\mathrm e}^{-4 c_{2}}}{6 x^{2} c_{3}^{4}}+c_{5} +\frac {4 x^{3}}{5} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {4 x^{3}}{5}-\frac {c_{1}}{24 x^{2}}+\frac {c_{2} x^{2}}{2}+c_{3} x +c_{4} \] |
Problem 9872
| ODE |
\[ \boxed {x y^{\prime \prime \prime \prime }-\left (6 x^{2}+1\right ) y^{\prime \prime \prime }+12 x^{3} y^{\prime \prime }-\left (9 x^{2}-7\right ) x^{2} y^{\prime }+2 \left (x^{2}-3\right ) x^{3} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -{\mathrm e}^{x^{2}} \left (\int \frac {\operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) c_{3} -{\mathrm e}^{x^{2}} \left (\int \frac {\operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) c_{4} +\left (\int \frac {\operatorname {WhittakerM}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) {\mathrm e}^{\frac {x^{2}}{2}} c_{3} +{\mathrm e}^{\frac {x^{2}}{2}} \left (\int \frac {\operatorname {WhittakerW}\left (\frac {9 \sqrt {5}}{20}, \frac {3}{4}, \frac {\sqrt {5}\, x^{2}}{2}\right ) {\mathrm e}^{\frac {x^{2}}{4}}}{x^{\frac {3}{2}}}d x \right ) c_{4} +c_{1} {\mathrm e}^{x^{2}}+c_{2} {\mathrm e}^{\frac {x^{2}}{2}} \] |
Problem 9873
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }-2 \left (\nu ^{2} x^{2}+6\right ) y^{\prime \prime }+\nu ^{2} \left (\nu ^{2} x^{2}+4\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\left (c_{4} \nu ^{2} x^{3}+6 c_{4} \nu \,x^{2}+15 c_{4} x +c_{2} \right ) {\mathrm e}^{-x \nu }+{\mathrm e}^{x \nu } \left (c_{3} \nu ^{2} x^{3}-6 c_{3} \nu \,x^{2}+15 c_{3} x +c_{1} \right )}{x} \] |
Problem 9874
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }+2 x y^{\prime \prime \prime }+a y=b \,x^{2}} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{4} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {-\sqrt {-a}}\, \sqrt {x}\right ) a +c_{3} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {-\sqrt {-a}}\, \sqrt {x}\right ) a +c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \left (-a \right )^{\frac {1}{4}} \sqrt {x}\right ) a +c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \left (-a \right )^{\frac {1}{4}} \sqrt {x}\right ) a +b \,x^{2}}{a} \] |
Problem 9875
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }+4 x y^{\prime \prime \prime }+2 y^{\prime \prime }=0} \] |
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program solution |
\[ y = c_{2} x \ln \left (x \right )-\ln \left (x \right ) c_{1} -c_{2} x +c_{3} x +c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{4} x +c_{2} \right ) \ln \left (x \right )+c_{3} x +c_{1} \] |
Problem 9876
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }+6 x y^{\prime \prime \prime }+6 y^{\prime \prime }=0} \] |
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program solution |
\[ y = c_{3} x -c_{2} \ln \left (x \right )+\frac {c_{1}}{2 x}+c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +\ln \left (x \right ) c_{2} +\frac {c_{3}}{x}+c_{4} x \] |
Problem 9877
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }+6 x y^{\prime \prime \prime }+6 y^{\prime \prime }-\lambda ^{2} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \operatorname {BesselJ}\left (1, 2 \sqrt {\lambda }\, \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (1, 2 \sqrt {\lambda }\, \sqrt {x}\right )+c_{4} \operatorname {BesselY}\left (1, 2 \sqrt {-\lambda }\, \sqrt {x}\right )+c_{3} \operatorname {BesselJ}\left (1, 2 \sqrt {-\lambda }\, \sqrt {x}\right )}{\sqrt {x}} \] |
Problem 9878
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }=0} \] |
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program solution |
\[ y = c_{3} x +\frac {c_{1}}{6 x^{2}}+\frac {c_{2}}{2 x}+c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +\frac {c_{2}}{x^{2}}+\frac {c_{3}}{x}+c_{4} x \] |
Problem 9879
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }-\lambda ^{2} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -\frac {-\operatorname {BesselJ}\left (1, 2 \sqrt {-\lambda }\, \sqrt {x}\right ) c_{3} \sqrt {\lambda }-\operatorname {BesselY}\left (1, 2 \sqrt {-\lambda }\, \sqrt {x}\right ) c_{4} \sqrt {\lambda }+\sqrt {-\lambda }\, \left (\sqrt {x}\, \sqrt {\lambda }\, \operatorname {BesselJ}\left (0, 2 \sqrt {-\lambda }\, \sqrt {x}\right ) c_{3} +\sqrt {x}\, \sqrt {\lambda }\, \operatorname {BesselY}\left (0, 2 \sqrt {-\lambda }\, \sqrt {x}\right ) c_{4} -c_{1} \operatorname {BesselJ}\left (1, 2 \sqrt {\lambda }\, \sqrt {x}\right )-c_{2} \operatorname {BesselY}\left (1, 2 \sqrt {\lambda }\, \sqrt {x}\right )+\sqrt {x}\, \sqrt {\lambda }\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {\lambda }\, \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (0, 2 \sqrt {\lambda }\, \sqrt {x}\right ) c_{2} \right )\right )}{\sqrt {-\lambda }\, \sqrt {\lambda }\, x^{\frac {3}{2}}} \] |
Problem 9880
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }+\left (2 n -2 \nu +4\right ) x y^{\prime \prime \prime }+\left (n -\nu +1\right ) \left (n -\nu +2\right ) y^{\prime \prime }-\frac {b^{4} y}{16}=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {BesselK}\left (n -\nu , b \sqrt {x}\right ) c_{3} +\operatorname {BesselI}\left (n -\nu , b \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (n -\nu , b \sqrt {x}\right ) c_{4} +\operatorname {BesselJ}\left (n -\nu , b \sqrt {x}\right ) c_{2} \right ) x^{-\frac {n}{2}+\frac {\nu }{2}} \] |
Problem 9881
| ODE |
\[ \boxed {x^{3} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime } x^{2}-y^{\prime \prime } x +y^{\prime }-a^{4} x^{3} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {BesselI}\left (0, a x \right )+c_{2} \operatorname {BesselJ}\left (0, a x \right )+c_{3} \operatorname {BesselK}\left (0, a x \right )+c_{4} \operatorname {BesselY}\left (0, a x \right ) \] |
Problem 9882
| ODE |
\[ \boxed {x^{3} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime } x^{2}+6 y^{\prime \prime } x=0} \] |
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program solution |
\[ y = c_{3} x -c_{2} \ln \left (x \right )+\frac {c_{1}}{2 x}+c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +\ln \left (x \right ) c_{2} +\frac {c_{3}}{x}+c_{4} x \] |
Problem 9883
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }-2 n \left (1+n \right ) x^{2} y^{\prime \prime }+4 n \left (1+n \right ) x y^{\prime }+\left (x^{4} a +n \left (1+n \right ) \left (n +3\right ) \left (-2+n \right )\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselY}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) c_{4} +\operatorname {BesselJ}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) c_{3} +\operatorname {BesselJ}\left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right ) c_{1} +\operatorname {BesselY}\left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right ) c_{2} \right ) \] |
Problem 9884
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime } x^{3}-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }+\left (4 n^{2}-1\right ) x y^{\prime }-4 y x^{4}=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {BesselY}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) c_{3} +\operatorname {BesselJ}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) c_{1} \right ) \operatorname {BesselJ}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )+\operatorname {BesselY}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right ) \left (\operatorname {BesselY}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) c_{4} +c_{2} \operatorname {BesselJ}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right )\right ) \] |
Problem 9885
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime } x^{3}-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }-\left (4 n^{2}-1\right ) x y^{\prime }+\left (-4 x^{4}+4 n^{2}-1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {n}{2}+\frac {1}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \frac {x^{4}}{64}\right )+\left (c_{3} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{2}, -\frac {n}{2}+1, \frac {n}{2}+1\right ], \frac {x^{4}}{64}\right )+c_{2} \operatorname {KelvinBei}\left (-n , x\right )^{2}+\operatorname {KelvinBer}\left (-n , x\right )^{2} c_{2} +c_{1} \left (\operatorname {KelvinBer}\left (n , x\right )^{2}+\operatorname {KelvinBei}\left (n , x\right )^{2}\right )\right ) x^{2}}{x} \] |
Problem 9886
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime } x^{3}-\left (4 n^{2}+3\right ) x^{2} y^{\prime \prime }+\left (12 n^{2}-3\right ) x y^{\prime }-\left (4 x^{4}+12 n^{2}-3\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{4} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {3}{2}-\frac {n}{2}, \frac {n}{2}+\frac {3}{2}\right ], \frac {x^{4}}{64}\right )+c_{3} x^{4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{2}, \frac {n}{2}+2, -\frac {n}{2}+2\right ], \frac {x^{4}}{64}\right )+c_{2} \operatorname {KelvinBei}\left (-n , x\right )^{2}+\operatorname {KelvinBer}\left (-n , x\right )^{2} c_{2} +c_{1} \left (\operatorname {KelvinBer}\left (n , x\right )^{2}+\operatorname {KelvinBei}\left (n , x\right )^{2}\right )}{x} \] |
Problem 9887
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime } x^{3}+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {BesselY}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right ) c_{2} +c_{1} \operatorname {BesselJ}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ) \operatorname {BesselJ}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right )+\operatorname {BesselY}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \left (\operatorname {BesselY}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right ) c_{4} +c_{3} \operatorname {BesselJ}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ) \] |
Problem 9888
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime } x^{3}+\left (4 x^{4}+\left (-2 \mu ^{2}-2 \nu ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-2 \mu ^{2}-2 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (8 x^{2}+\left (\mu ^{2}-\nu ^{2}\right )^{2}\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {BesselY}\left (\mu , x\right ) c_{2} +c_{1} \operatorname {BesselJ}\left (\mu , x\right )\right ) \operatorname {BesselJ}\left (\nu , x\right )+\operatorname {BesselY}\left (\nu , x\right ) \left (\operatorname {BesselY}\left (\mu , x\right ) c_{4} +c_{3} \operatorname {BesselJ}\left (\mu , x\right )\right ) \] |
Problem 9889
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime } x^{3}+12 x^{2} y^{\prime \prime }=0} \] |
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program solution |
\[ y = \frac {c_{1}}{x}+c_{2} +\frac {c_{3}}{x^{2}}+c_{4} x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} +\frac {c_{2}}{x^{2}}+\frac {c_{3}}{x}+c_{4} x \] |
Problem 9890
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime } x^{3}+12 x^{2} y^{\prime \prime }+a y=0} \] |
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program solution |
\[ y = c_{1} x^{-\frac {1}{2}+\frac {\sqrt {5-4 \sqrt {1-a}}}{2}}+c_{2} x^{-\frac {1}{2}+\frac {\sqrt {5+4 \sqrt {1-a}}}{2}}+c_{3} x^{-\frac {1}{2}-\frac {\sqrt {5-4 \sqrt {1-a}}}{2}}+c_{4} x^{-\frac {1}{2}-\frac {\sqrt {5+4 \sqrt {1-a}}}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} x^{-\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}+c_{2} x^{\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}+c_{3} x^{-\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}+c_{4} x^{\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}}{\sqrt {x}} \] |
Problem 9891
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a \right ) x^{3} y^{\prime \prime \prime }+\left (4 x^{2 c} b^{2} c^{2}+6 \left (-1+a \right )^{2}-2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )+1\right ) x^{2} y^{\prime \prime }+\left (4 \left (3 c -2 a +1\right ) b^{2} c^{2} x^{2 c}+\left (2 a -1\right ) \left (2 c^{2} \left (\mu ^{2}+\nu ^{2}\right )-2 a \left (-1+a \right )-1\right )\right ) x y^{\prime }+\left (4 \left (a -c \right ) \left (a -2 c \right ) b^{2} c^{2} x^{2 c}+\left (c \mu +c \nu +a \right ) \left (c \mu +c \nu -a \right ) \left (c \mu -c \nu +a \right ) \left (c \mu -c \nu -a \right )\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x^{a} \left (\left (\operatorname {BesselJ}\left (\mu , b \,x^{c}\right ) c_{1} +\operatorname {BesselY}\left (\mu , b \,x^{c}\right ) c_{3} \right ) \operatorname {BesselJ}\left (\nu , b \,x^{c}\right )+\operatorname {BesselY}\left (\nu , b \,x^{c}\right ) \left (c_{4} \operatorname {BesselY}\left (\mu , b \,x^{c}\right )+\operatorname {BesselJ}\left (\mu , b \,x^{c}\right ) c_{2} \right )\right ) \] |
Problem 9892
| ODE |
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a -4 c \right ) x^{3} y^{\prime \prime \prime }+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (a +c -1\right )^{2}+4 \left (-1+a \right ) \left (c -1\right )-1\right ) x^{2} y^{\prime \prime }+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (2 a -1\right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 c a +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (\nu , b \,x^{c}\right ) c_{1} +\operatorname {BesselY}\left (\nu , b \,x^{c}\right ) c_{2} +\operatorname {BesselY}\left (\nu , i b \,x^{c}\right ) c_{4} +\operatorname {BesselJ}\left (\nu , i b \,x^{c}\right ) c_{3} \right ) x^{a} \] |
Problem 9893
| ODE |
\[ \boxed {\nu ^{4} x^{4} y^{\prime \prime \prime \prime }+\left (-2+4 \nu \right ) \nu ^{3} x^{3} y^{\prime \prime \prime }+\left (\nu -1\right ) \left (2 \nu -1\right ) \nu ^{2} x^{2} y^{\prime \prime }-\frac {b^{4} x^{\frac {2}{\nu }} y}{16}=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_{1} +\operatorname {BesselY}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {-\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_{4} +\operatorname {BesselJ}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {-\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_{3} \right ) \] |
Problem 9894
| ODE |
\[ \boxed {\left (x^{2}-1\right )^{2} y^{\prime \prime \prime \prime }+10 x \left (x^{2}-1\right ) y^{\prime \prime \prime }+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (1+\nu \right )\right ) \left (x^{2}-1\right )\right ) y^{\prime \prime }-6 x \left (\mu \left (\mu +1\right )+\nu \left (1+\nu \right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (1+\nu \right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (1+\nu \right )\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {LegendreQ}\left (\mu , x\right ) c_{2} +c_{1} \operatorname {LegendreP}\left (\mu , x\right )\right ) \operatorname {LegendreP}\left (\nu , x\right )+\operatorname {LegendreQ}\left (\nu , x\right ) \left (\operatorname {LegendreQ}\left (\mu , x\right ) c_{4} +c_{3} \operatorname {LegendreP}\left (\mu , x\right )\right ) \] |
Problem 9895
| ODE |
\[ \boxed {\left (2 x +{\mathrm e}^{x}\right ) y^{\prime \prime \prime \prime }+4 \left ({\mathrm e}^{x}+2\right ) y^{\prime \prime \prime }+6 y^{\prime \prime } {\mathrm e}^{x}+4 y^{\prime } {\mathrm e}^{x}+y \,{\mathrm e}^{x}=\frac {1}{x^{5}}} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {24 c_{1} x^{4}+24 c_{2} x^{3}+24 x^{2} c_{3} +24 c_{4} x +1}{24 \left ({\mathrm e}^{x}+2 x \right ) x} \] |
Problem 9896
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime } \sin \left (x \right )^{4}+2 y^{\prime \prime \prime } \sin \left (x \right )^{3} \cos \left (x \right )+y^{\prime \prime } \sin \left (x \right )^{2} \left (\sin \left (x \right )^{2}-3\right )+y^{\prime } \sin \left (x \right ) \cos \left (x \right ) \left (2 \sin \left (x \right )^{2}+3\right )+\left (a^{4} \sin \left (x \right )^{4}-3\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \sin \left (x \right ) \left (c_{1} \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )+c_{2} \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )+\cos \left (x \right ) \left (\operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right ) c_{3} +c_{4} \operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right )\right )\right ) \] |
Problem 9897
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime } \sin \left (x \right )^{6}+4 y^{\prime \prime \prime } \sin \left (x \right )^{5} \cos \left (x \right )-6 y^{\prime \prime } \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}=f} \] |
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program solution |
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 9898
| ODE |
\[ \boxed {f \left (y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y\right )+2 \operatorname {df} \left (y^{\prime \prime \prime }-a^{2} y^{\prime }\right )=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {\left (\operatorname {df} +\sqrt {a^{2} f^{2}+\operatorname {df}^{2}}\right ) x}{f}} c_{1} +{\mathrm e}^{\frac {\left (-\operatorname {df} +\sqrt {a^{2} f^{2}+\operatorname {df}^{2}}\right ) x}{f}} c_{2} +{\mathrm e}^{x a} c_{3} +{\mathrm e}^{-x a} c_{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{a x}+c_{2} {\mathrm e}^{-a x}+c_{3} {\mathrm e}^{\frac {\left (-\operatorname {df} +\sqrt {a^{2} f^{2}+\operatorname {df}^{2}}\right ) x}{f}}+c_{4} {\mathrm e}^{-\frac {\left (\operatorname {df} +\sqrt {a^{2} f^{2}+\operatorname {df}^{2}}\right ) x}{f}} \] |
Problem 9899
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime } f=0} \] |
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program solution |
\[ y = c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{6} c_{1} x^{3}+\frac {1}{2} c_{2} x^{2}+c_{3} x +c_{4} \] |
Problem 9900
| ODE |
\[ \boxed {y^{\prime \prime \prime \prime }-2 a^{2} y^{\prime \prime }+a^{4} y-\lambda \left (x a -b \right ) \left (y^{\prime \prime }-a^{2} y\right )=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{a x} \left (\int {\mathrm e}^{-2 a x} \left (\int {\mathrm e}^{a x} \left (c_{3} \operatorname {AiryAi}\left (-\frac {\left (-a \lambda \right )^{\frac {1}{3}} \left (\lambda \left (a x -b \right )+a^{2}\right )}{a \lambda }\right )+c_{4} \operatorname {AiryBi}\left (-\frac {\left (-a \lambda \right )^{\frac {1}{3}} \left (\lambda \left (a x -b \right )+a^{2}\right )}{a \lambda }\right )\right )d x +c_{2} \right )d x +c_{1} \right ) \] |