Problem 10801
| ODE |
\[ \boxed {y y^{\prime }+\frac {a \left (\frac {\left (3 n +5\right ) x}{2}+\frac {n -1}{n +1}\right ) x^{-\frac {n +4}{n +3}} y}{n +3}=-\frac {a^{2} \left (\left (n +1\right ) x^{2}-\frac {\left (n^{2}+2 n +5\right ) x}{n +1}+\frac {4}{n +1}\right ) x^{-\frac {5+n}{n +3}}}{2 n +6}} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10802
| ODE |
\[ \boxed {y y^{\prime }-a \left (\frac {n +2}{n}+b \,x^{n}\right ) y=-\frac {a^{2} x \left (\frac {n +1}{n}+b \,x^{n}\right )}{n}} \] |
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program solution |
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Maple solution | \[ -n \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, \left (\int _{}^{\frac {2 \arctan \left (\frac {2 x^{n +1} a b n +\left (n +1\right ) \left (a x -y \left (x \right ) n \right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n \left (a x -y \left (x \right ) n \right )}\right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}}\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right )+\left (-2 b n \,x^{n}-n -1\right ) {\mathrm e}^{-\frac {2 \arctan \left (\frac {2 x^{n +1} a b n +\left (n +1\right ) \left (a x -y \left (x \right ) n \right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n \left (a x -y \left (x \right ) n \right )}\right )}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}}+c_{1} = 0 \] |
Problem 10803
| ODE |
\[ \boxed {y y^{\prime }-\left (a \,{\mathrm e}^{x}+b \right ) y=c \,{\mathrm e}^{2 x}-a b \,{\mathrm e}^{x}-b^{2}} \] |
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program solution |
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Maple solution | \[ \sqrt {\frac {c \,{\mathrm e}^{2 x}-\left (b -y \left (x \right )\right ) \left (a \,{\mathrm e}^{x}+b -y \left (x \right )\right )}{\left (b -y \left (x \right )\right )^{2}}}\, y \left (x \right ) {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {\left (b -y \left (x \right )\right ) a -2 \,{\mathrm e}^{x} c}{\sqrt {a^{2}+4 c}\, \left (b -y \left (x \right )\right )}\right )}{\sqrt {a^{2}+4 c}}}-b \left (\int _{}^{\frac {{\mathrm e}^{x}}{-b +y \left (x \right )}}\frac {\sqrt {\textit {\_a}^{2} c +a \textit {\_a} -1}\, {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {2 c \textit {\_a} +a}{\sqrt {a^{2}+4 c}}\right )}{\sqrt {a^{2}+4 c}}}}{\textit {\_a}}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 10804
| ODE |
\[ \boxed {y y^{\prime }-\left (a \left (2 \mu +\lambda \right ) {\mathrm e}^{\lambda x}+b \right ) {\mathrm e}^{\mu x} y=\left (-a^{2} \mu \,{\mathrm e}^{2 \lambda x}-a b \,{\mathrm e}^{\lambda x}+c \right ) {\mathrm e}^{2 \mu x}} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10805
| ODE |
\[ \boxed {y y^{\prime }-\left (a \,{\mathrm e}^{\lambda x}+b \right ) y=c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right )} \] |
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program solution |
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Maple solution | \[ \frac {\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, \left (\frac {c \lambda }{2}+\frac {1}{6}\right ) \ln \left (\frac {\left (3 c \lambda +1\right )^{2} \left (b^{2} c \,\lambda ^{2} x^{2}+2 \,{\mathrm e}^{x \lambda } a b c \lambda x +{\mathrm e}^{2 x \lambda } a^{2} c +b \lambda x y \left (x \right )+a \,{\mathrm e}^{x \lambda } y \left (x \right )-\lambda y \left (x \right )^{2}\right ) c}{\left (9 c \lambda +2\right ) y \left (x \right )^{2}}\right )-3 \left (c \lambda +\frac {1}{3}\right )^{2} \operatorname {arctanh}\left (\frac {\left (3 c \lambda +1\right ) \left (2 b c \lambda x +2 \,{\mathrm e}^{x \lambda } a c +y \left (x \right )\right )}{\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, y \left (x \right )}\right )+\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, \left (\left (-c \lambda -\frac {1}{3}\right ) \ln \left (\frac {\left (3 c \lambda +1\right ) \left (b \lambda x +{\mathrm e}^{x \lambda } a \right ) c}{y \left (x \right )}\right )+\left (c \lambda +\frac {1}{3}\right ) \ln \left (b \lambda x +{\mathrm e}^{x \lambda } a \right )-c_{1} c \lambda \right )}{\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, c \lambda } = 0 \] |
Problem 10806
| ODE |
\[ \boxed {y y^{\prime }-{\mathrm e}^{\lambda x} \left (2 a x \lambda +a +b \right ) y=-{\mathrm e}^{2 \lambda x} \left (a^{2} \lambda \,x^{2}+a b x +c \right )} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\left (\tan \left (\frac {\operatorname {RootOf}\left (2 a x \lambda \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}\, \tan \left (\frac {\textit {\_a} \sqrt {-\frac {b^{2}-4 c \lambda }{a^{2}}}}{2}\right ) \textit {\_Z} a +b \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_{1} a \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}}{2}\right ) a \sqrt {\frac {-b^{2}+4 c \lambda }{a^{2}}}+2 x a \lambda +b \right ) {\mathrm e}^{x \lambda }}{2 \lambda } \] |
Problem 10807
| ODE |
\[ \boxed {y y^{\prime }-{\mathrm e}^{a x} \left (2 a \,x^{2}+b +2 x \right ) y={\mathrm e}^{2 a x} \left (-x^{4} a -b \,x^{2}+c \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10808
| ODE |
\[ \boxed {y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y=-a^{2} b \,x^{2} {\mathrm e}^{2 b x}} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -\frac {{\mathrm e}^{b x} a \left (b x \operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right )+c_{1} \right )-1\right )}{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b x -\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right )+c_{1} \right ) b} \] |
Problem 10809
| ODE |
\[ \boxed {y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y=-a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x}} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{\left (n +1\right ) x} a \left (1+2 x \,n^{2}+\left (\tan \left (\frac {\operatorname {RootOf}\left (2 x \,n^{2} {\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) \textit {\_Z} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n +2 n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_{1} n \,{\mathrm e}^{\textit {\_a}}+{\mathrm e}^{\textit {\_Z} +\textit {\_a}}\right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}+2 x +1\right ) n \right )}{2 n +2} \] |
Problem 10810
| ODE |
\[ \boxed {y y^{\prime }+a \left (1+2 b \sqrt {x}\right ) {\mathrm e}^{2 b \sqrt {x}} y=-a^{2} b \,x^{\frac {3}{2}} {\mathrm e}^{4 b \sqrt {x}}} \] |
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program solution |
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Maple solution | \[ \frac {\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, \sqrt {x}\, \operatorname {BesselI}\left (1, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) c_{1} b -\operatorname {BesselK}\left (1, -\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, b \sqrt {x}-\operatorname {BesselI}\left (0, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) c_{1} +\operatorname {BesselK}\left (0, -\sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right )}{\operatorname {BesselI}\left (1, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\, b \sqrt {x}-\operatorname {BesselI}\left (0, \sqrt {\frac {a \,{\mathrm e}^{2 b \sqrt {x}}}{b^{2} \left ({\mathrm e}^{2 b \sqrt {x}} a x +y \left (x \right )\right )}}\right )} = 0 \] |
Problem 10811
| ODE |
\[ \boxed {y y^{\prime }-\left (a \cosh \left (x \right )+b \right ) y=-a b \sinh \left (x \right )+c} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10812
| ODE |
\[ \boxed {y y^{\prime }-\left (a \sinh \left (x \right )+b \right ) y=-a b \cosh \left (x \right )+c} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10813
| ODE |
\[ \boxed {y y^{\prime }-\left (2 \ln \left (x \right )+a +1\right ) y=x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right )} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {x \left (-\tanh \left (\frac {\operatorname {RootOf}\left (-\sqrt {a^{2}+4 b}\, \tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} \tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \ln \left (x \right ) {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} a -{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} a +2 c_{1} \right ) \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \ln \left (x \right )+a \right )}{2} \] |
Problem 10814
| ODE |
\[ \boxed {y y^{\prime }-\left (2 \ln \left (x \right )^{2}+2 \ln \left (x \right )+a \right ) y=x \left (-\ln \left (x \right )^{4}-\ln \left (x \right )^{2} a +b \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10815
| ODE |
\[ \boxed {y y^{\prime }-a x \cos \left (\lambda \,x^{2}\right ) y=x} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10816
| ODE |
\[ \boxed {y y^{\prime }-a x \sin \left (\lambda \,x^{2}\right ) y=x} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10817
| ODE |
\[ \boxed {\left (A y+B x +a \right ) y^{\prime }+B y=-x k -b} \] |
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program solution |
\[ \frac {k \,x^{2}}{2}+\left (B y+b \right ) x +\frac {A y^{2}}{2}+a y = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\sqrt {-\left (A k -B^{2}\right ) \left (\left (k x +b \right ) A -B^{2} x -B a \right )^{2} c_{1}^{2}+A}+\left (k \left (-B x -a \right ) A +x \,B^{3}+a \,B^{2}\right ) c_{1}}{A c_{1} \left (A k -B^{2}\right )} \] |
Problem 10818
| ODE |
\[ \boxed {\left (y+a x +b \right ) y^{\prime }-\alpha y=\beta x +\gamma } \] |
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program solution |
\[ \frac {\frac {\ln \left (\left (-a \alpha \beta +\beta ^{2}\right ) x^{2}+\left (\left (\left (-a +\alpha \right ) y+2 \gamma +\left (-\alpha -a \right ) b \right ) \beta +a \left (\alpha y+\gamma \right ) \left (a -\alpha \right )\right ) x -\left (b +y\right )^{2} \beta +\left (a y+\gamma +b \left (a -\alpha \right )\right ) \left (\alpha y+\gamma \right )\right ) \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }}{2}+\arctan \left (\frac {\left (a -\alpha \right ) \left (a \alpha -\beta \right ) x +\left (b \alpha +2 \alpha y+\gamma \right ) a +\left (-2 b -2 y\right ) \beta -\alpha ^{2} b +\alpha \gamma }{\sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }\, \left (\left (a \alpha -\beta \right ) x +b \alpha -\gamma \right )}\right ) \left (\alpha +a \right )}{\sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (\left (a x +b \right ) \alpha -x \beta -\gamma \right ) \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }\, \tan \left (\operatorname {RootOf}\left (-2 \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }\, \ln \left (2\right )+\sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }\, \ln \left (-\sec \left (\textit {\_Z} \right )^{2} \left (a^{2}-2 a \alpha +\alpha ^{2}+4 \beta \right ) \left (a \alpha x +b \alpha -x \beta -\gamma \right )^{2}\right )+2 c_{1} \sqrt {-a^{2}+2 a \alpha -\alpha ^{2}-4 \beta }+2 \textit {\_Z} a +2 \textit {\_Z} \alpha \right )\right )+\left (a x +b \right ) \alpha ^{2}+\left (-a^{2} x -a b -x \beta -\gamma \right ) \alpha +\left (x \beta -\gamma \right ) a +2 b \beta }{2 a \alpha -2 \beta } \] |
Problem 10819
| ODE |
\[ \boxed {\left (y+a k \,x^{2}+b x +c \right ) y^{\prime }+a y^{2}-2 y a k x -y m=k \left (k +b -m \right ) x +s} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10820
| ODE |
\[ \boxed {\left (y+A \,x^{n}+a \right ) y^{\prime }+n A \,x^{n -1} y=-k \,x^{m}-b} \] |
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program solution |
\[ b x +\frac {k \,x^{1+m}}{1+m}+y A \,x^{n}+a y+\frac {y^{2}}{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {\left (-2 x^{1+m} k +\left (1+m \right ) \left (x^{2 n} A^{2}+2 A \,x^{n} a +a^{2}-2 b x -2 c_{1} \right )\right ) \left (1+m \right )}+A \left (-m -1\right ) x^{n}-a m -a}{1+m} \\ y \left (x \right ) &= \frac {-\sqrt {\left (-2 x^{1+m} k +\left (1+m \right ) \left (x^{2 n} A^{2}+2 A \,x^{n} a +a^{2}-2 b x -2 c_{1} \right )\right ) \left (1+m \right )}+A \left (-m -1\right ) x^{n}-a m -a}{1+m} \\ \end{align*} |
Problem 10821
| ODE |
\[ \boxed {\left (y+a \,x^{n +1}+b \,x^{n}\right ) y^{\prime }-\left (a n \,x^{n}+c \,x^{n -1}\right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10822
| ODE |
\[ \boxed {y y^{\prime } x -a y^{2}-y b=c \,x^{n}+s} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10823
| ODE |
\[ \boxed {y y^{\prime } x +y^{2} n -a \left (1+2 n \right ) x y-y b=-n \,a^{2} x^{2}-a b x +c} \] |
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program solution |
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Maple solution | \[ \frac {\left (\frac {-n y \left (x \right )^{2}+\left (2 a x n +b \right ) y \left (x \right )-a^{2} n \,x^{2}-a b x +c}{\left (a x -y \left (x \right )\right )^{2}}\right )^{-\frac {1}{2 n}} \left (\frac {1}{a x -y \left (x \right )}\right )^{\frac {1}{n}} y \left (x \right ) {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-a b x +b y \left (x \right )+2 c}{\sqrt {b^{2}+4 c n}\, \left (-a x +y \left (x \right )\right )}\right )}{\sqrt {b^{2}+4 c n}\, n}}-\left (\left (\int _{}^{\frac {1}{a x -y \left (x \right )}}\left (\textit {\_a}^{2} c -\textit {\_a} b -n \right )^{-\frac {1}{2 n}} {\mathrm e}^{\frac {b \,\operatorname {arctanh}\left (\frac {-2 c \textit {\_a} +b}{\sqrt {b^{2}+4 c n}}\right )}{n \sqrt {b^{2}+4 c n}}} \textit {\_a}^{\frac {1}{n}}d \textit {\_a} \right ) a -c_{1} \right ) \left (a x -y \left (x \right )\right ) x}{x \left (a x -y \left (x \right )\right )} = 0 \] |
Problem 10824
| ODE |
\[ \boxed {y^{\prime \prime }+a y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{\sqrt {-a}\, x}-\frac {c_{2} {\mathrm e}^{-\sqrt {-a}\, x}}{2 \sqrt {-a}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sin \left (\sqrt {a}\, x \right )+c_{2} \cos \left (\sqrt {a}\, x \right ) \] |
Problem 10825
| ODE |
\[ \boxed {y^{\prime \prime }-\left (a x +b \right ) y=0} \] |
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program solution |
\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {AiryAi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right )+c_{2} \operatorname {AiryBi}\left (\frac {a x +b}{\left (-a \right )^{\frac {2}{3}}}\right ) \] |
Problem 10826
| ODE |
\[ \boxed {y^{\prime \prime }-\left (a^{2} x^{2}+a \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{\frac {a \,x^{2}}{2}}+\frac {c_{2} {\mathrm e}^{\frac {a \,x^{2}}{2}} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {a}\, x \right )}{2 \sqrt {a}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{\frac {a \,x^{2}}{2}} \left (c_{1} +\operatorname {erf}\left (\sqrt {a}\, x \right ) c_{2} \right ) \] |
Problem 10827
| ODE |
\[ \boxed {y^{\prime \prime }-\left (a \,x^{2}+b \right ) y=0} \] |
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program solution |
\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \operatorname {WhittakerM}\left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {b}{4 \sqrt {a}}, \frac {1}{4}, \sqrt {a}\, x^{2}\right )}{\sqrt {x}} \] |
Problem 10828
| ODE |
\[ \boxed {y^{\prime \prime }+a^{3} x \left (-a x +2\right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {a x \left (a x -2\right )}{2}}-\frac {i c_{2} {\mathrm e}^{-\frac {1}{2} a^{2} x^{2}+a x -1} \sqrt {\pi }\, \operatorname {erf}\left (i a x -i\right )}{2 a} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {a x \left (a x -2\right )}{2}} \left (c_{1} +\operatorname {erf}\left (i a x -i\right ) c_{2} \right ) \] |
Problem 10829
| ODE |
\[ \boxed {y^{\prime \prime }-\left (a \,x^{2}+b c x \right ) y=0} \] |
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program solution |
\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (a x +b c \right )}{2 \sqrt {a}}} \left (2 x a c_{2} \operatorname {hypergeom}\left (\left [-\frac {b^{2} c^{2}-12 a^{\frac {3}{2}}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (2 a x +b c \right )^{2}}{4 a^{\frac {3}{2}}}\right )+c b c_{2} \operatorname {hypergeom}\left (\left [-\frac {b^{2} c^{2}-12 a^{\frac {3}{2}}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (2 a x +b c \right )^{2}}{4 a^{\frac {3}{2}}}\right )+\operatorname {hypergeom}\left (\left [-\frac {b^{2} c^{2}-4 a^{\frac {3}{2}}}{16 a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (2 a x +b c \right )^{2}}{4 a^{\frac {3}{2}}}\right ) c_{1} \right ) \] |
Problem 10830
| ODE |
\[ \boxed {y^{\prime \prime }-x^{n} y a=0} \] |
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program solution |
\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{\frac {n}{2}+1}}{n +2}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{\frac {n}{2}+1}}{n +2}\right ) c_{1} \right ) \] |
Problem 10831
| ODE |
\[ \boxed {y^{\prime \prime }-a \left (x^{2 n} a +x^{n -1} n \right ) y=0} \] |
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program solution |
\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{2} x^{-\frac {3 n}{2}-1} \left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {2 a \,x^{n +1}}{n +1}\right )}{2}+\left (\left (\frac {n}{2}+1\right ) x^{-\frac {3 n}{2}-1}+a \,x^{-\frac {n}{2}}\right ) \left (n +1\right ) c_{2} \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {2 a \,x^{n +1}}{n +1}\right )+c_{1} {\mathrm e}^{\frac {a \,x^{n +1}}{n +1}} \] |
Problem 10832
| ODE |
\[ \boxed {y^{\prime \prime }-a \,x^{-2+n} \left (a \,x^{n}+n +1\right ) y=0} \] |
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program solution |
\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{2} x^{-\frac {3 n}{2}+\frac {1}{2}} \left (n -1\right )^{2} \operatorname {WhittakerM}\left (\frac {n -1}{2 n}, \frac {2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )}{2}+\left (\frac {\left (n -1\right ) x^{-\frac {3 n}{2}+\frac {1}{2}}}{2}+x^{-\frac {n}{2}+\frac {1}{2}} a \right ) n c_{2} \operatorname {WhittakerM}\left (-\frac {n +1}{2 n}, \frac {2 n -1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+c_{1} x \,{\mathrm e}^{\frac {a \,x^{n}}{n}} \] |
Problem 10833
| ODE |
\[ \boxed {y^{\prime \prime }+\left (x^{2 n} a +b \,x^{n -1}\right ) y=0} \] |
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program solution |
\[ y = -c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (1, 2 \sqrt {x}\right )-c_{2} \sqrt {x}\, \operatorname {BesselY}\left (1, 2 \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{-\frac {n}{2}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {i b}{\sqrt {a}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, x \,x^{n}}{n +1}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i b}{\sqrt {a}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, x \,x^{n}}{n +1}\right )\right ) \] |
Problem 10834
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a +y b=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}}-\frac {c_{2} {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}}}{\sqrt {a^{2}-4 b}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) x}{2}}+c_{2} {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) x}{2}} \] |
Problem 10835
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a +\left (b x +c \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {a x}{2}} \left (\operatorname {AiryAi}\left (\frac {a^{2}-4 b x -4 c}{4 b^{\frac {2}{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (\frac {a^{2}-4 b x -4 c}{4 b^{\frac {2}{3}}}\right ) c_{2} \right ) \] |
Problem 10836
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a -\left (b \,x^{2}+c \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (\sqrt {b}\, x +a \right )}{2}} x \left (\operatorname {KummerM}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) c_{2} \right ) \] |
Problem 10837
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a +b \left (-b \,x^{2}+a x +1\right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {b \,x^{2}}{2}}+\frac {c_{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {2 b^{2} x^{2}+a^{2}}{4 b}} \operatorname {erf}\left (\frac {-2 b x +a}{2 \sqrt {-b}}\right )}{2 \sqrt {-b}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{2} b}{2}} \left (c_{1} \operatorname {erf}\left (\frac {-2 b x +a}{2 \sqrt {-b}}\right )+c_{2} \right ) \] |
Problem 10838
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a +b x \left (-x^{3} b +a x +2\right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {x^{3} b}{3}}+c_{2} {\mathrm e}^{-\frac {x^{3} b}{3}} \left (\int {\mathrm e}^{\frac {2}{3} x^{3} b -a x}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\left (\int {\mathrm e}^{-a x +\frac {2}{3} x^{3} b}d x \right ) c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {x^{3} b}{3}} \] |
Problem 10839
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a +b \left (-b \,x^{2 n}+a \,x^{n}+x^{n -1} n \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\left (\int {\mathrm e}^{\frac {2 b \,x^{n +1}-x a \left (n +1\right )}{n +1}}d x \right ) c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {b \,x^{n +1}}{n +1}} \] |
Problem 10840
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a +b \left (-b \,x^{2 n}-a \,x^{n}+x^{n -1} n \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (b \,x^{n}+a \left (n +1\right )\right )}{n +1}} \left (c_{1} +\left (\int {\mathrm e}^{\frac {x \left (2 b \,x^{n}+a \left (n +1\right )\right )}{n +1}}d x \right ) c_{2} \right ) \] |
Problem 10841
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } x +\left (n -1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -\frac {\left (-2 \left (-\frac {x^{2}}{2}+n +\frac {1}{2}\right ) c_{1} n \operatorname {KummerM}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+2 \left (-x^{2}+2 n +1\right ) c_{2} \operatorname {KummerU}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+n c_{1} \left (n +2\right ) \operatorname {KummerM}\left (-\frac {n}{2}-\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right )+4 \operatorname {KummerU}\left (-\frac {n}{2}-\frac {1}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) c_{2} \right ) x \,{\mathrm e}^{-\frac {x^{2}}{2}}}{n \left (n -1\right )} \] |
Problem 10842
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime } x +2 n y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x \left (\operatorname {KummerU}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, x^{2}\right ) c_{2} +\operatorname {KummerM}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, x^{2}\right ) c_{1} \right ) \] |
Problem 10843
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a x +y b=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {a \,x^{2}}{2}} x \left (\operatorname {KummerM}\left (\frac {-b +2 a}{2 a}, \frac {3}{2}, \frac {a \,x^{2}}{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {-b +2 a}{2 a}, \frac {3}{2}, \frac {a \,x^{2}}{2}\right ) c_{2} \right ) \] |
Problem 10844
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a x +y b x=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {b x}{a}} \left (\operatorname {KummerM}\left (\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x -2 b \right )^{2}}{2 a^{3}}\right ) c_{1} +\operatorname {KummerU}\left (\frac {b^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x -2 b \right )^{2}}{2 a^{3}}\right ) c_{2} \right ) \] |
Problem 10845
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } a x +\left (b x +c \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {b x}{a}} \left (\operatorname {KummerM}\left (\frac {a^{2} c +b^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x -2 b \right )^{2}}{2 a^{3}}\right ) c_{1} +\operatorname {KummerU}\left (\frac {a^{2} c +b^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x -2 b \right )^{2}}{2 a^{3}}\right ) c_{2} \right ) \] |
Problem 10846
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime } a x +\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {\left (i \sqrt {b}\, x +\frac {3 a}{2}\right ) x^{2}}{3}} x \left (\operatorname {KummerM}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right ) c_{1} +\operatorname {KummerU}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right ) c_{2} \right ) \] |
Problem 10847
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }-a y=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (a x +b \right )}{a}-\frac {c_{2} \left (\sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{2 a}} \left (a x +b \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +b \right )}{2 \sqrt {a}}\right )+2 \,{\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \sqrt {a}\right )}{2 \sqrt {a}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{\frac {b^{2}}{2 a}} \pi c_{2} \left (a x +b \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +b \right )}{2 \sqrt {a}}\right )+\sqrt {\pi }\, \sqrt {2}\, \sqrt {a}\, {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} c_{2} +c_{1} \left (a x +b \right ) \] |
Problem 10848
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+a y=0} \] |
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program solution |
\[ y = -\frac {{\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} c_{1} \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{2 a}} \operatorname {erf}\left (-\frac {\sqrt {-2 a}\, x}{2}+\frac {b}{\sqrt {-2 a}}\right )}{\sqrt {-2 a}}+c_{2} {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {erf}\left (\frac {\left (a x +b \right ) \sqrt {2}}{2 \sqrt {-a}}\right ) c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \] |
Problem 10849
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (a x +b -c \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-c x}+\frac {c_{2} \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {-2 a c x +\left (b -2 c \right )^{2}}{2 a}} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +b -2 c \right )}{2 \sqrt {a}}\right )}{2 \sqrt {a}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-c x} \left (c_{1} +\operatorname {erf}\left (\frac {\sqrt {2}\, \left (a x +b -2 c \right )}{2 \sqrt {a}}\right ) c_{2} \right ) \] |
Problem 10850
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a x +2 b \right ) y^{\prime }+\left (a b x +b^{2}-a \right ) y=0} \] |
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program solution |
\[ y = c_{1} x \,{\mathrm e}^{-b x}-\frac {c_{2} {\mathrm e}^{-b x} \left (\sqrt {a}\, \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) x +2 \,{\mathrm e}^{-\frac {a \,x^{2}}{2}}\right )}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 \,{\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} c_{2} +{\mathrm e}^{-b x} x \left (c_{2} \sqrt {a}\, \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )+c_{1} \right ) \] |
Problem 10851
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {c x}{a}} \left (\operatorname {KummerM}\left (\frac {d \,a^{2}-a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right ) c_{1} +\operatorname {KummerU}\left (\frac {d \,a^{2}-a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (a^{2} x +a b -2 c \right )^{2}}{2 a^{3}}\right ) c_{2} \right ) \] |
Problem 10852
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {c \,x^{2}}{2}} \left (c_{1} +\operatorname {erf}\left (\frac {\left (-2 c +a \right ) x +b}{\sqrt {2 a -4 c}}\right ) c_{2} \right ) \] |
Problem 10853
| ODE |
\[ \boxed {y^{\prime \prime }+2 \left (a x +b \right ) y^{\prime }+\left (a^{2} x^{2}+2 a b x +c \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {x \left (a x -2 \sqrt {b^{2}+a -c}+2 b \right )}{2}}-\frac {c_{2} {\mathrm e}^{-\frac {x \left (a x +2 \sqrt {b^{2}+a -c}+2 b \right )}{2}}}{2 \sqrt {b^{2}+a -c}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \left (a x -2 \sqrt {b^{2}+a -c}+2 b \right )}{2}}+c_{2} {\mathrm e}^{-\frac {x \left (a x +2 \sqrt {b^{2}+a -c}+2 b \right )}{2}} \] |
Problem 10854
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (\left (a x +2 b \right ) \sqrt {a^{2}-4 \alpha }+x \left (a^{2}-4 \alpha \right )+2 a b -4 \beta \right )}{4 \sqrt {a^{2}-4 \alpha }}} \left (c_{2} \left (a^{2} x +a b -4 \alpha x -2 \beta \right ) \operatorname {hypergeom}\left (\left [\frac {3 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \gamma +2 \left (b \beta -2 \alpha \right ) a +2 \left (-b^{2}+4 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (a^{2} x +a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right )+\operatorname {hypergeom}\left (\left [\frac {\left (a^{2}-4 \alpha \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \gamma +\left (2 b \beta -4 \alpha \right ) a +\left (-2 b^{2}+8 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (a^{2} x +a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{\frac {3}{2}}}\right ) c_{1} \right ) \] |
Problem 10855
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (-x^{2 n} c +a \,x^{n +1}+b \,x^{n}+x^{n -1} n \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10856
| ODE |
\[ \boxed {y^{\prime \prime }+a \left (-b^{2}+x^{2}\right ) y^{\prime }-a \left (b +x \right ) y=0} \] |
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program solution |
\[ y = c_{1} \left (x -b \right )-c_{2} \left (-x +b \right ) \left (\int \frac {{\mathrm e}^{a \,b^{2} x -\frac {1}{3} a \,x^{3}}}{\left (-x +b \right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {HeunT}\left (-\frac {a 3^{\frac {2}{3}} b}{\left (a^{2}\right )^{\frac {1}{3}}}, -6 \,\operatorname {csgn}\left (a \right ), -\frac {a^{2} b^{2} 3^{\frac {1}{3}}}{\left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{\frac {x \left (3 b^{2}-x^{2}\right ) \operatorname {csgn}\left (a \right ) a \left (\operatorname {csgn}\left (a \right )+1\right )}{6}}+c_{2} \operatorname {HeunT}\left (-\frac {a 3^{\frac {2}{3}} b}{\left (a^{2}\right )^{\frac {1}{3}}}, 6 \,\operatorname {csgn}\left (a \right ), -\frac {a^{2} b^{2} 3^{\frac {1}{3}}}{\left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{\frac {x \left (3 b^{2}-x^{2}\right ) \operatorname {csgn}\left (a \right ) a \left (\operatorname {csgn}\left (a \right )-1\right )}{6}} \] |
Problem 10857
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+c \left (a \,x^{2}+b -c \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-c x}+c_{2} {\mathrm e}^{-c x} \left (\int {\mathrm e}^{-\frac {x \left (a \,x^{2}+3 b -6 c \right )}{3}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {HeunT}\left (0, -3 \,\operatorname {csgn}\left (a \right ), \frac {a \left (b -2 c \right ) 3^{\frac {1}{3}}}{\left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{-\frac {x \left (\left (a \,x^{2}+3 b \right ) \operatorname {csgn}\left (a \right )+a \,x^{2}+3 b -6 c \right ) \operatorname {csgn}\left (a \right )}{6}}+c_{2} \operatorname {HeunT}\left (0, 3 \,\operatorname {csgn}\left (a \right ), \frac {a \left (b -2 c \right ) 3^{\frac {1}{3}}}{\left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} \left (a^{2}\right )^{\frac {1}{6}} x}{3}\right ) {\mathrm e}^{-\frac {x \left (\left (a \,x^{2}+3 b \right ) \operatorname {csgn}\left (a \right )-a \,x^{2}-3 b +6 c \right ) \operatorname {csgn}\left (a \right )}{6}} \] |
Problem 10858
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+2 b \right ) y^{\prime }+\left (a b \,x^{2}-a x +b^{2}\right ) y=0} \] |
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program solution |
\[ y = c_{1} x \,{\mathrm e}^{-b x}+c_{2} x \,{\mathrm e}^{-b x} \left (\int \frac {{\mathrm e}^{-\frac {a \,x^{3}}{3}}}{x^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {5 \left (3^{\frac {2}{3}} c_{2} a \left (a \,x^{3}\right )^{\frac {1}{3}} \left (a \,x^{3}+2\right ) {\mathrm e}^{-\frac {x \left (a \,x^{2}+6 b \right )}{6}}+\frac {9 x^{2} \left (c_{2} a^{2} x \,{\mathrm e}^{-b x} \operatorname {WhittakerM}\left (\frac {1}{3}, \frac {5}{6}, \frac {a \,x^{3}}{3}\right )+c_{1} {\mathrm e}^{\frac {x \left (a \,x^{2}-6 b \right )}{6}}\right )}{5}\right ) {\mathrm e}^{-\frac {a \,x^{3}}{6}}}{9 x} \] |
Problem 10859
| ODE |
\[ \boxed {y^{\prime \prime }+\left (2 x^{2}+a \right ) y^{\prime }+\left (x^{4}+a \,x^{2}+b +2 x \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {x^{3}}{3}-\frac {a x}{2}+\frac {\sqrt {a^{2}-4 b}\, x}{2}}-\frac {c_{2} {\mathrm e}^{-\frac {x \left (2 x^{2}+3 \sqrt {a^{2}-4 b}+3 a \right )}{6}}}{\sqrt {a^{2}-4 b}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x \left (-2 x^{2}+3 \sqrt {a^{2}-4 b}-3 a \right )}{6}}+c_{2} {\mathrm e}^{-\frac {x \left (2 x^{2}+3 \sqrt {a^{2}-4 b}+3 a \right )}{6}} \] |
Problem 10860
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {\operatorname {csgn}\left (a \right ) x \left (2 a^{2} x^{2} \operatorname {csgn}\left (a \right )+3 a b x \,\operatorname {csgn}\left (a \right )+2 a^{2} x^{2}+3 a b x -12 \alpha \right )}{12 a}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{\frac {1}{3}}}, -\frac {3 \left (a^{2}-\beta a +b \alpha \right ) \operatorname {csgn}\left (a \right )}{a^{2}}, -\frac {3^{\frac {1}{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {\operatorname {csgn}\left (a \right ) x \left (2 a^{2} x^{2} \operatorname {csgn}\left (a \right )+3 a b x \,\operatorname {csgn}\left (a \right )-2 a^{2} x^{2}-3 a b x +12 \alpha \right )}{12 a}} \operatorname {HeunT}\left (\frac {3^{\frac {2}{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{\frac {1}{3}}}, \frac {3 \left (a^{2}-\beta a +b \alpha \right ) \operatorname {csgn}\left (a \right )}{a^{2}}, -\frac {3^{\frac {1}{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right ) \] |
Problem 10861
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a b \,x^{2}+b x +2 a \right ) y^{\prime }+a^{2} \left (b \,x^{2}+1\right ) y=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (a x +1\right ) {\mathrm e}^{-a x}}{a}+c_{2} \left (a x +1\right ) {\mathrm e}^{-a x} a \left (\int \frac {{\mathrm e}^{-\frac {x^{2} \left (a x +\frac {3}{2}\right ) b}{3}}}{\left (a x +1\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \left (2 a^{2} b^{2} x^{2}+2 a b \,x^{2} \sqrt {a^{2} b^{2}}+3 a \,b^{2} x +3 b x \sqrt {a^{2} b^{2}}+12 a \sqrt {a^{2} b^{2}}\right )}{12 \sqrt {a^{2} b^{2}}}} \operatorname {HeunT}\left (\frac {b 3^{\frac {2}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}, -\frac {6 a b}{\sqrt {a^{2} b^{2}}}, -\frac {b^{2} 3^{\frac {1}{3}}}{4 \left (a^{2} b^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \,b^{2} \left (2 a x +1\right )}{6 \left (a^{2} b^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {\left (-2 a^{2} b^{2} x^{2}+2 a b \,x^{2} \sqrt {a^{2} b^{2}}-3 a \,b^{2} x +3 b x \sqrt {a^{2} b^{2}}+12 a \sqrt {a^{2} b^{2}}\right ) x}{12 \sqrt {a^{2} b^{2}}}} \operatorname {HeunT}\left (\frac {b 3^{\frac {2}{3}}}{2 \left (a^{2} b^{2}\right )^{\frac {1}{3}}}, \frac {6 a b}{\sqrt {a^{2} b^{2}}}, -\frac {b^{2} 3^{\frac {1}{3}}}{4 \left (a^{2} b^{2}\right )^{\frac {2}{3}}}, -\frac {\left (a x +\frac {1}{2}\right ) 3^{\frac {2}{3}} b^{2} a}{3 \left (a^{2} b^{2}\right )^{\frac {5}{6}}}\right ) \] |
Problem 10862
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+x \left (a b \,x^{2}+b c +2 a \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {1}{3} a \,x^{3}-c x}+c_{2} {\mathrm e}^{-\frac {x \left (a \,x^{2}+3 c \right )}{3}} \left (\int {\mathrm e}^{\frac {1}{3} a \,x^{3}-\frac {1}{2} b \,x^{2}+c x}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \,\operatorname {csgn}\left (a \right ) \left (\left (a \,x^{2}+\frac {3}{2} b x +3 c \right ) \operatorname {csgn}\left (a \right )+a \,x^{2}-\frac {3 b x}{2}+3 c \right )}{6}} \operatorname {HeunT}\left (0, 3 \,\operatorname {csgn}\left (a \right ), \frac {3^{\frac {1}{3}} \left (4 a c -b^{2}\right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 a x -b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {x \left (\left (a \,x^{2}+\frac {3}{2} b x +3 c \right ) \operatorname {csgn}\left (a \right )-a \,x^{2}+\frac {3 b x}{2}-3 c \right ) \operatorname {csgn}\left (a \right )}{6}} \operatorname {HeunT}\left (0, -3 \,\operatorname {csgn}\left (a \right ), \frac {3^{\frac {1}{3}} \left (4 a c -b^{2}\right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} \left (a x -\frac {b}{2}\right ) a}{3 \left (a^{2}\right )^{\frac {5}{6}}}\right ) \] |
Problem 10863
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (a b \,x^{3}+a c \,x^{2}+b \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {x \left (b x +2 c \right )}{2}}+c_{2} {\mathrm e}^{-\frac {x \left (b x +2 c \right )}{2}} \left (\int {\mathrm e}^{-\frac {x \left (a \,x^{2}-\frac {3}{2} b x -3 c \right )}{3}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \,\operatorname {csgn}\left (a \right ) \left (\left (a \,x^{2}+\frac {3}{2} b x +3 c \right ) \operatorname {csgn}\left (a \right )+a \,x^{2}-\frac {3 b x}{2}-3 c \right )}{6}} \operatorname {HeunT}\left (0, -3 \,\operatorname {csgn}\left (a \right ), -\frac {3^{\frac {1}{3}} \left (4 a c +b^{2}\right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, \frac {3^{\frac {2}{3}} a \left (2 a x -b \right )}{6 \left (a^{2}\right )^{\frac {5}{6}}}\right )+c_{2} {\mathrm e}^{-\frac {x \,\operatorname {csgn}\left (a \right ) \left (\left (a \,x^{2}+\frac {3}{2} b x +3 c \right ) \operatorname {csgn}\left (a \right )-a \,x^{2}+\frac {3 b x}{2}+3 c \right )}{6}} \operatorname {HeunT}\left (0, 3 \,\operatorname {csgn}\left (a \right ), -\frac {3^{\frac {1}{3}} \left (4 a c +b^{2}\right )}{4 \left (a^{2}\right )^{\frac {2}{3}}}, -\frac {3^{\frac {2}{3}} \left (a x -\frac {b}{2}\right ) a}{3 \left (a^{2}\right )^{\frac {5}{6}}}\right ) \] |
Problem 10864
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a b \,x^{3}-a \,x^{2}+b^{2}\right ) y=0} \] |
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program solution |
\[ y = c_{1} x \,{\mathrm e}^{-b x}+c_{2} x \,{\mathrm e}^{-b x} \left (\int \frac {{\mathrm e}^{-\frac {x^{4} a}{4}}}{x^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\frac {7 \,2^{\frac {1}{4}} c_{2} a \left (x^{4} a \right )^{\frac {3}{8}} \left (x^{4} a +3\right ) {\mathrm e}^{-\frac {x \left (a \,x^{3}+4 b \right )}{4}}}{8}+{\mathrm e}^{-\frac {x \left (a \,x^{3}+8 b \right )}{8}} \operatorname {WhittakerM}\left (\frac {3}{8}, \frac {7}{8}, \frac {x^{4} a}{4}\right ) c_{2} a^{2} x^{4}+{\mathrm e}^{-b x} c_{1} x^{\frac {5}{2}}}{x^{\frac {3}{2}}} \] |
Problem 10865
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 a \,x^{2}+b \right ) y=0} \] |
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program solution |
\[ y = c_{3} {\mathrm e}^{\int -\frac {\left (\int c_{1} {\mathrm e}^{\frac {x^{4} a}{4}+\frac {b \,x^{2}}{2}-2 \ln \left (x \right )}d x \right ) a \,x^{4}+c_{2} a \,x^{4}+\left (\int c_{1} {\mathrm e}^{\frac {x^{4} a}{4}+\frac {b \,x^{2}}{2}-2 \ln \left (x \right )}d x \right ) b \,x^{2}+c_{2} b \,x^{2}-c_{1} {\mathrm e}^{\frac {x^{4} a}{4}+\frac {b \,x^{2}}{2}-2 \ln \left (x \right )} x -\left (\int c_{1} {\mathrm e}^{\frac {x^{4} a}{4}+\frac {b \,x^{2}}{2}-2 \ln \left (x \right )}d x \right )-c_{2}}{\left (\int c_{1} {\mathrm e}^{\frac {x^{4} a}{4}+\frac {b \,x^{2}}{2}-2 \ln \left (x \right )}d x +c_{2} \right ) x}d x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {\left (a \,x^{2}+2 b \right ) x^{2}}{4}} \left (\operatorname {HeunB}\left (\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right ) c_{1} x +\operatorname {HeunB}\left (-\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right ) c_{2} \right ) \] |
Problem 10866
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+2 a \right ) y^{\prime }+a^{2} \left (x^{3} b +1\right ) y=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (a x +1\right ) {\mathrm e}^{-a x}}{a}+c_{2} \left (a x +1\right ) {\mathrm e}^{-a x} a \left (\int \frac {{\mathrm e}^{-\frac {x^{3} \left (a x +\frac {4}{3}\right ) b}{4}}}{\left (a x +1\right )^{2}}d x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-a x} \left (c_{2} \left (\int \frac {{\mathrm e}^{-\frac {b \,x^{3} \left (a x +\frac {4}{3}\right )}{4}}}{\left (a x +1\right )^{2}}d x \right )+c_{1} \right ) \left (a x +1\right ) \] |
Problem 10867
| ODE |
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }=0} \] |
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program solution |
\[ y = \frac {c_{1} \left (\frac {a}{n +1}\right )^{-\frac {1}{n +1}} \left (\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-n -1} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (\frac {a \,x^{n +1} n^{2}}{n +1}+\frac {2 a \,x^{n +1} n}{n +1}+n^{2}+\frac {a \,x^{n +1}}{n +1}+3 n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}, \frac {n +2}{2+2 n}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (n +2\right ) \left (2 n +3\right ) a}+\frac {\left (n +1\right )^{2} x^{\frac {n}{n +1}+\frac {1}{n +1}-n -1} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (n +2\right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {n +2}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {1}{n +1}-\frac {n +2}{2 \left (n +1\right )}+1, \frac {n +2}{2+2 n}+\frac {1}{2}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (2 n +3\right ) a}\right )}{n +1}+c_{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{-n} \left (\left (\frac {a x \,x^{n}}{n +1}\right )^{\frac {-n -2}{2 n +2}} c_{2} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (n +2\right )^{2} \left (n +1\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a x \,x^{n}}{n +1}\right )+\left (\frac {a x \,x^{n}}{n +1}\right )^{\frac {-n -2}{2 n +2}} c_{2} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {x^{n} a x}{2 n +2}} \left (n +1\right )^{3} \left (x^{n} a x +n +2\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, \frac {a x \,x^{n}}{n +1}\right )+2 c_{1} \left (n +2\right ) a \left (n +\frac {3}{2}\right ) x^{n}\right )}{\left (n +2\right ) \left (2 n +3\right ) a} \] |
Problem 10868
| ODE |
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }+y x^{n -1} b=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x \left (\operatorname {KummerU}\left (\frac {1+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right ) c_{2} +\operatorname {KummerM}\left (\frac {1+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right ) c_{1} \right ) {\mathrm e}^{-\frac {a x \,x^{n}}{n +1}} \] |
Problem 10869
| ODE |
\[ \boxed {y^{\prime \prime }+2 a \,x^{n} y^{\prime }+a \left (x^{2 n} a +x^{n -1} n \right ) y=0} \] |
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program solution |
\[ y = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}} \left (c_{2} x +c_{1} \right ) \] |
Problem 10870
| ODE |
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{n +1} \left (a +\sqrt {a^{2}-4 b}\right )}{2 n +2}} x \left (\operatorname {KummerM}\left (\frac {\left (n +2\right ) \sqrt {a^{2}-4 b}+a n -2 c}{\sqrt {a^{2}-4 b}\, \left (2 n +2\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x^{n +1}}{n +1}\right ) c_{1} +\operatorname {KummerU}\left (\frac {\left (n +2\right ) \sqrt {a^{2}-4 b}+a n -2 c}{\sqrt {a^{2}-4 b}\, \left (2 n +2\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x^{n +1}}{n +1}\right ) c_{2} \right ) \] |
Problem 10871
| ODE |
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }-b \left (a \,x^{n +m}+b \,x^{2 m}+x^{m -1} m \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10872
| ODE |
\[ \boxed {y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+a n \,x^{n -1}+c \,x^{m -1}\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x \left (\operatorname {KummerM}\left (\frac {\left (m +2\right ) \sqrt {b}+i c}{\sqrt {b}\, \left (2 m +2\right )}, \frac {m +2}{1+m}, \frac {2 i \sqrt {b}\, x^{1+m}}{1+m}\right ) c_{1} +\operatorname {KummerU}\left (\frac {\left (m +2\right ) \sqrt {b}+i c}{\sqrt {b}\, \left (2 m +2\right )}, \frac {m +2}{1+m}, \frac {2 i \sqrt {b}\, x^{1+m}}{1+m}\right ) c_{2} \right ) {\mathrm e}^{\frac {-i \left (n +1\right ) \sqrt {b}\, x^{1+m}-x^{n +1} a \left (1+m \right )}{\left (n +1\right ) \left (1+m \right )}} \] |
Problem 10873
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+c \left (a \,x^{n}+b -c \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10874
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (x^{n} a b -a \,x^{n -1}+b^{2}\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {\left (a \,x^{n}+2 \left (n +1\right ) b \right ) x}{2 n +2}} c_{2} \left (n +1\right ) \left (a \,x^{-\frac {n}{2}}+x^{-\frac {3 n}{2}-1} n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, \frac {a \,x^{n +1}}{n +1}\right )+c_{2} n^{2} x^{-\frac {3 n}{2}-1} {\mathrm e}^{-\frac {\left (a \,x^{n}+2 \left (n +1\right ) b \right ) x}{2 n +2}} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, \frac {a \,x^{n +1}}{n +1}\right )+c_{1} {\mathrm e}^{-b x} x \] |
Problem 10875
| ODE |
\[ \boxed {y^{\prime \prime }+\left (x^{n} a b +b \,x^{n -1}+2 a \right ) y^{\prime }+a^{2} \left (b \,x^{n}+1\right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10876
| ODE |
\[ \boxed {y^{\prime \prime }+\left (x^{n} a b +2 b \,x^{n -1}-a^{2} x \right ) y^{\prime }+a \left (x^{n} a b +b \,x^{n -1}-a^{2} x \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10877
| ODE |
\[ \boxed {y^{\prime \prime }+x^{n} \left (a \,x^{2}+\left (a c +b \right ) x +b c \right ) y^{\prime }-x^{n} \left (a x +b \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -\left (x +c \right ) \left (\left (\int \frac {{\mathrm e}^{-\frac {\left (a \,x^{2} \left (n +2\right ) \left (n +1\right )+\left (a c +b \right ) x \left (3+n \right ) \left (n +1\right )+b c \left (3+n \right ) \left (n +2\right )\right ) x^{n +1}}{\left (3+n \right ) \left (n +1\right ) \left (n +2\right )}}}{\left (x +c \right )^{2}}d x \right ) c_{1} +c_{2} \right ) \] |
Problem 10878
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }-\left (a \,x^{n -1}+b \,x^{m -1}\right ) y=0} \] |
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program solution |
\[ y = \left (\int \frac {c_{1} {\mathrm e}^{-\frac {b \,x^{1+m}}{1+m}} {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}}}{x^{2}}d x +c_{2} \right ) x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (c_{1} +c_{2} \left (\int \frac {{\mathrm e}^{-\frac {\left (b \left (n +1\right ) x^{m}+a \left (1+m \right ) x^{n}\right ) x}{\left (1+m \right ) \left (n +1\right )}}}{x^{2}}d x \right )\right ) \] |
Problem 10879
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {\left (b \,x^{m} \left (n +1\right )+a \,x^{n} \left (1+m \right )\right ) x}{\left (1+m \right ) \left (n +1\right )}} \left (c_{1} \left (\int {\mathrm e}^{\frac {\left (b \,x^{m} \left (n +1\right )+a \,x^{n} \left (1+m \right )\right ) x}{\left (1+m \right ) \left (n +1\right )}}d x \right )+c_{2} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{1} \left (\int {\mathrm e}^{\frac {\left (b \left (n +1\right ) x^{m}+a \left (1+m \right ) x^{n}\right ) x}{\left (1+m \right ) \left (n +1\right )}}d x \right )+c_{2} \right ) {\mathrm e}^{-\frac {\left (b \left (n +1\right ) x^{m}+a \left (1+m \right ) x^{n}\right ) x}{\left (1+m \right ) \left (n +1\right )}} \] |
Problem 10880
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (a \left (n +1\right ) x^{n -1}+b \left (1+m \right ) x^{m -1}\right ) y=0} \] |
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program solution |
\[ y = c_{3} {\mathrm e}^{\int -\frac {a \,x^{n} x^{2} \left (\int \frac {c_{1} {\mathrm e}^{\frac {b x \,x^{m}}{1+m}} {\mathrm e}^{\frac {a x \,x^{n}}{n +1}}}{x^{2}}d x \right )+a \,x^{n} x^{2} c_{2} +b \,x^{m} x^{2} \left (\int \frac {c_{1} {\mathrm e}^{\frac {b x \,x^{m}}{1+m}} {\mathrm e}^{\frac {a x \,x^{n}}{n +1}}}{x^{2}}d x \right )+b \,x^{m} x^{2} c_{2} -c_{1} {\mathrm e}^{\frac {b x \,x^{m}}{1+m}} {\mathrm e}^{\frac {a x \,x^{n}}{n +1}}-\left (\int \frac {c_{1} {\mathrm e}^{\frac {b x \,x^{m}}{1+m}} {\mathrm e}^{\frac {a x \,x^{n}}{n +1}}}{x^{2}}d x \right ) x -c_{2} x}{x^{2} \left (\int \frac {c_{1} {\mathrm e}^{\frac {b x \,x^{m}}{1+m}} {\mathrm e}^{\frac {a x \,x^{n}}{n +1}}}{x^{2}}d x +c_{2} \right )}d x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (c_{1} +\left (\int \frac {{\mathrm e}^{\frac {\left (b \left (n +1\right ) x^{m}+a \left (1+m \right ) x^{n}\right ) x}{\left (1+m \right ) \left (n +1\right )}}}{x^{2}}d x \right ) c_{2} \right ) {\mathrm e}^{-\frac {\left (b \left (n +1\right ) x^{m}+a \left (1+m \right ) x^{n}\right ) x}{\left (1+m \right ) \left (n +1\right )}} \] |
Problem 10881
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+c \left (a \,x^{n}+b \,x^{m}-c \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10882
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }+\left (x^{n +m} a b +b \left (1+m \right ) x^{m -1}-a \,x^{n -1}\right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10883
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (x^{n +m} a b +x^{m} b c +a n \,x^{n -1}\right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10884
| ODE |
\[ \boxed {x y^{\prime \prime }+\frac {y^{\prime }}{2}+a y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{2 \sqrt {-a x}}-\frac {c_{2} \sqrt {-a x}\, \left ({\mathrm e}^{2 \sqrt {-a x}}-{\mathrm e}^{-2 \sqrt {-a x}}\right )}{2 a \sqrt {x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sin \left (2 \sqrt {x}\, \sqrt {a}\right )+c_{2} \cos \left (2 \sqrt {x}\, \sqrt {a}\right ) \] |
Problem 10885
| ODE |
\[ \boxed {x y^{\prime \prime }+y^{\prime } a +y b=0} \] |
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program solution |
\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (1-a , 2 \sqrt {b}\, \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (1-a , 2 \sqrt {b}\, \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (-\sqrt {x}\, \operatorname {BesselJ}\left (a +1, 2 \sqrt {b}\, \sqrt {x}\right ) \sqrt {b}\, c_{1} -\sqrt {x}\, \operatorname {BesselY}\left (a +1, 2 \sqrt {b}\, \sqrt {x}\right ) \sqrt {b}\, c_{2} +a \left (\operatorname {BesselJ}\left (a , 2 \sqrt {b}\, \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (a , 2 \sqrt {b}\, \sqrt {x}\right ) c_{2} \right )\right ) x^{-\frac {a}{2}}}{\sqrt {b}} \] |
Problem 10886
| ODE |
\[ \boxed {x y^{\prime \prime }+y^{\prime } a +y b x=0} \] |
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program solution |
\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (\frac {1}{2}-\frac {a}{2}, x \sqrt {b}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (\frac {1}{2}-\frac {a}{2}, x \sqrt {b}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (\frac {a}{2}-\frac {1}{2}, \sqrt {b}\, x \right ) c_{1} +\operatorname {BesselY}\left (\frac {a}{2}-\frac {1}{2}, \sqrt {b}\, x \right ) c_{2} \right ) x^{-\frac {a}{2}+\frac {1}{2}} \] |
Problem 10887
| ODE |
\[ \boxed {x y^{\prime \prime }+y^{\prime } a +\left (b x +c \right ) y=0} \] |
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program solution |
\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (1-a , 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (1-a , 2 \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-i \sqrt {b}\, x} \left (\operatorname {KummerU}\left (\frac {i c +a \sqrt {b}}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i c +a \sqrt {b}}{2 \sqrt {b}}, a , 2 i \sqrt {b}\, x \right ) c_{1} \right ) \] |
Problem 10888
| ODE |
\[ \boxed {x y^{\prime \prime }+n y^{\prime }+b \,x^{1-2 n} y=0} \] |
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program solution |
\[ y = c_{1} \cos \left (\frac {x^{1-n} \sqrt {b}}{n -1}\right )-c_{2} \sin \left (\frac {x^{1-n} \sqrt {b}}{n -1}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sin \left (\frac {x^{-n +1} \sqrt {b}}{n -1}\right )+c_{2} \cos \left (\frac {x^{-n +1} \sqrt {b}}{n -1}\right ) \] |
Problem 10889
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{2} {\mathrm e}^{-a \,x^{n}} \left (a \,x^{n}+x^{-n} \sqrt {x^{2 n}}\right )-{\mathrm e}^{a \,x^{n}} c_{1} \left (a \,x^{n}-x^{-n} \sqrt {x^{2 n}}\right ) \] |
Problem 10890
| ODE |
\[ \boxed {x y^{\prime \prime }+y^{\prime } a +b \,x^{n} y=0} \] |
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program solution |
\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (-\frac {a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (-\frac {a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\operatorname {BesselY}\left (\frac {a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) c_{1} \right ) x^{-\frac {a}{2}+\frac {1}{2}} \] |
Problem 10891
| ODE |
\[ \boxed {x y^{\prime \prime }+y^{\prime } a +b \,x^{n} \left (-b \,x^{n +1}+a +n \right ) y=0} \] |
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program solution |
\[ y = c_{1} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselJ}\left (1-a , 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {a}{2}} \operatorname {BesselY}\left (1-a , 2 \sqrt {x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\left (\left (a -n -2\right ) x^{-\frac {3 n}{2}-\frac {a}{2}-1}+2 b \,x^{-\frac {n}{2}-\frac {a}{2}}\right ) c_{2} \left (n +1\right ) \operatorname {WhittakerM}\left (\frac {-a -n}{2 n +2}, \frac {-a +2 n +3}{2 n +2}, -\frac {2 b \,x^{n +1}}{n +1}\right )+x^{-\frac {3 n}{2}-\frac {a}{2}-1} c_{2} \left (a -n -2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2-a}{2 n +2}, \frac {-a +2 n +3}{2 n +2}, -\frac {2 b \,x^{n +1}}{n +1}\right )+c_{1} {\mathrm e}^{-\frac {b \,x^{n +1}}{n +1}} \] |
Problem 10892
| ODE |
\[ \boxed {x y^{\prime \prime }+y^{\prime } a x +a y=0} \] |
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program solution |
\[ y = -\operatorname {expIntegral}_{1}\left (-a x \right ) c_{1} a x \,{\mathrm e}^{-a x}-c_{1} +c_{2} x \,{\mathrm e}^{-a x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \operatorname {expIntegral}_{1}\left (-a x \right ) c_{1} a x \,{\mathrm e}^{-a x}+c_{1} +c_{2} x \,{\mathrm e}^{-a x} \] |
Problem 10893
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (-x +b \right ) y^{\prime }-a y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {KummerM}\left (a , b , x\right )+c_{2} \operatorname {KummerU}\left (a , b , x\right ) \] |
Problem 10894
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{-c x}+c_{2} x^{-\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {b}{2}, \frac {1}{2}-\frac {b}{2}, \left (-2 c +a \right ) x \right ) {\mathrm e}^{-\frac {a x}{2}} \] |
Problem 10895
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+a \left (a x +b \right ) y=0} \] |
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program solution |
\[ y = c_{1} x^{-b +1} {\mathrm e}^{-a x}+\frac {c_{2} {\mathrm e}^{-a x}}{b -1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-a x} \left (c_{1} +x^{-b +1} c_{2} \right ) \] |
Problem 10896
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (\left (a +b \right ) x +n +m \right ) y^{\prime }+\left (a b x +n a +b m \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-a x} \left (\operatorname {KummerU}\left (m , n +m , \left (a -b \right ) x \right ) c_{2} +\operatorname {KummerM}\left (m , n +m , \left (a -b \right ) x \right ) c_{1} \right ) \] |
Problem 10897
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x \left (\sqrt {a^{2}-4 c}+a \right )}{2}} \left (\operatorname {KummerM}\left (\frac {b \sqrt {a^{2}-4 c}+a b -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {b \sqrt {a^{2}-4 c}+a b -2 d}{2 \sqrt {a^{2}-4 c}}, b , \sqrt {a^{2}-4 c}\, x \right ) c_{2} \right ) \] |
Problem 10898
| ODE |
\[ \boxed {x y^{\prime \prime }-\left (a x +1\right ) y^{\prime }-b \,x^{2} \left (b x +a \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {b \,x^{2}}{2}}-\frac {c_{2} {\mathrm e}^{-\frac {b \,x^{2}}{2}} \left (a \sqrt {\pi }\, {\mathrm e}^{-\frac {a^{2}}{4 b}} \operatorname {erf}\left (\frac {2 b x +a}{2 \sqrt {-b}}\right )+2 \,{\mathrm e}^{x \left (b x +a \right )} \sqrt {-b}\right )}{4 \left (-b \right )^{\frac {3}{2}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -{\mathrm e}^{-\frac {2 b^{2} x^{2}+a^{2}}{4 b}} \sqrt {\pi }\, \sqrt {-b}\, \operatorname {erf}\left (\frac {2 b x +a}{2 \sqrt {-b}}\right ) c_{2} a +2 \,{\mathrm e}^{\frac {1}{2} x^{2} b +a x} c_{2} b +c_{1} {\mathrm e}^{-\frac {x^{2} b}{2}} \] |
Problem 10899
| ODE |
\[ \boxed {x y^{\prime \prime }-\left (2 a x +1\right ) y^{\prime }+\left (x^{3} b +a^{2} x +a \right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{a x -\frac {i \sqrt {b}\, x^{2}}{2}}-\frac {i c_{2} {\mathrm e}^{\frac {x \left (i \sqrt {b}\, x +2 a \right )}{2}}}{2 \sqrt {b}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{a x +\frac {x^{2} \sqrt {-b}}{2}}+c_{2} {\mathrm e}^{a x -\frac {x^{2} \sqrt {-b}}{2}} \] |
Problem 10900
| ODE |
\[ \boxed {x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }=-c x \left (-c \,x^{2}+a x +b +1\right )} \] |
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program solution |
\[ y = \int -\frac {-\left (-a x \right )^{-b} b^{3} c^{2} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-3 \left (-a x \right )^{-b} b^{2} c^{2} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-2 b \,c^{2} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) {\mathrm e}^{-a x}-x^{-b} c_{1} a^{3} {\mathrm e}^{-a x}-a^{2} c^{2} x^{2}+c^{2} \left (-a x \right )^{-b} \Gamma \left (b +3\right ) {\mathrm e}^{-a x}+a^{3} c x +a \,c^{2} b x +2 a \,c^{2} x -c^{2} b^{2}-3 b \,c^{2}-2 c^{2}}{a^{3}}d x +c_{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{2} a^{3}-\left (\int \left (-c^{2} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) {\mathrm e}^{-a x} \left (-a x \right )^{-b}-{\mathrm e}^{-a x} x^{-b} c_{1} a^{3}+\left (\left (-b^{2}+\left (a x -3\right ) b -a^{2} x^{2}+2 a x -2\right ) c +a^{3} x \right ) c \right )d x \right )}{a^{3}} \] |