Problem 10324
| ODE |
\[ \boxed {y^{\prime }=f \left (x \right )} \] |
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program solution |
\[ y = \int f \left (x \right )d x +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \int f \left (x \right )d x +c_{1} \] |
Problem 10325
| ODE |
\[ \boxed {y^{\prime }-f \left (y\right )=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{f \left (\textit {\_a} \right )}d \textit {\_a} = x +c_{1} \] Verified OK.
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Maple solution | \[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{f \left (\textit {\_a} \right )}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 10326
| ODE |
\[ \boxed {y^{\prime }-f \left (x \right ) g \left (y\right )=0} \] |
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program solution |
\[ \int _{}^{x}-f \left (\textit {\_a} \right )d \textit {\_a} +\int _{0}^{y}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} = c_{1} \] Verified OK.
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Maple solution | \[ \int f \left (x \right )d x -\left (\int _{}^{y \left (x \right )}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 10327
| ODE |
\[ \boxed {g \left (x \right ) y^{\prime }-f_{1} \left (x \right ) y=f_{0} \left (x \right )} \] |
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program solution |
\[ \int _{}^{x}-\left (f_{1} \left (\textit {\_a} \right ) y+f_{0} \left (\textit {\_a} \right )\right ) {\mathrm e}^{-\left (\int \frac {f_{1} \left (\textit {\_a} \right )+\frac {d}{d \textit {\_a}}g \left (\textit {\_a} \right )}{g \left (\textit {\_a} \right )}d \textit {\_a} \right )}d \textit {\_a} +\left (g \left (x \right ) {\mathrm e}^{-\left (\int \frac {f_{1} \left (x \right )+g^{\prime }\left (x \right )}{g \left (x \right )}d x \right )}+\int _{}^{x}f_{1} \left (\textit {\_a} \right ) {\mathrm e}^{-\left (\int \frac {f_{1} \left (\textit {\_a} \right )+\frac {d}{d \textit {\_a}}g \left (\textit {\_a} \right )}{g \left (\textit {\_a} \right )}d \textit {\_a} \right )}d \textit {\_a} \right ) y = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\int \frac {f_{0} \left (x \right ) {\mathrm e}^{-\left (\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x \right )}}{g \left (x \right )}d x +c_{1} \right ) {\mathrm e}^{\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x} \] |
Problem 10328
| ODE |
\[ \boxed {g \left (x \right ) y^{\prime }-f_{1} \left (x \right ) y-f_{n} \left (x \right ) y^{n}=0} \] |
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program solution |
\[ y^{1-n} = -\left (\left (-1+n \right ) \left (\int \frac {f_{n} \left (x \right ) {\mathrm e}^{\left (-1+n \right ) \left (\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x \right )}}{g \left (x \right )}d x \right )-c_{1} \right ) {\mathrm e}^{-\left (-1+n \right ) \left (\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x} {\left (-n \left (\int \frac {f_{n} \left (x \right ) {\mathrm e}^{\left (n -1\right ) \left (\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x \right )}}{g \left (x \right )}d x \right )+c_{1} +\int \frac {f_{n} \left (x \right ) {\mathrm e}^{\left (n -1\right ) \left (\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x \right )}}{g \left (x \right )}d x \right )}^{-\frac {1}{n -1}} \] |
Problem 10329
| ODE |
\[ \boxed {y^{\prime }-f \left (\frac {y}{x}\right )=0} \] |
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program solution |
\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x}}\frac {1}{f \left (\textit {\_a} \right )-\textit {\_a}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}-\frac {1}{-f \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} \right )+\ln \left (x \right )+c_{1} \right ) x \] |
Problem 10330
| ODE |
\[ \boxed {-a y^{2}+y^{\prime }=b x +c} \] |
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program solution |
\[ y = \frac {\left (\operatorname {AiryAi}\left (1, -\frac {\left (a b \right )^{\frac {1}{3}} \left (b x +c \right )}{b}\right ) c_{3} +\operatorname {AiryBi}\left (1, -\frac {\left (a b \right )^{\frac {1}{3}} \left (b x +c \right )}{b}\right )\right ) \left (a b \right )^{\frac {1}{3}}}{a \left (c_{3} \operatorname {AiryAi}\left (-\frac {\left (a b \right )^{\frac {1}{3}} \left (b x +c \right )}{b}\right )+\operatorname {AiryBi}\left (-\frac {\left (a b \right )^{\frac {1}{3}} \left (b x +c \right )}{b}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (\frac {b}{\sqrt {a}}\right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\frac {b x +c}{\left (\frac {b}{\sqrt {a}}\right )^{\frac {2}{3}}}\right ) c_{1} +\operatorname {AiryBi}\left (1, -\frac {b x +c}{\left (\frac {b}{\sqrt {a}}\right )^{\frac {2}{3}}}\right )\right )}{\sqrt {a}\, \left (c_{1} \operatorname {AiryAi}\left (-\frac {b x +c}{\left (\frac {b}{\sqrt {a}}\right )^{\frac {2}{3}}}\right )+\operatorname {AiryBi}\left (-\frac {b x +c}{\left (\frac {b}{\sqrt {a}}\right )^{\frac {2}{3}}}\right )\right )} \] |
Problem 10331
| ODE |
\[ \boxed {y^{\prime }-y^{2}=-x^{2} a^{2}+3 a} \] |
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program solution |
\[ y = \frac {x a \,{\mathrm e}^{x^{2} a}-\sqrt {\pi }\, \left (x^{2} \left (-a \right )^{\frac {3}{2}}+\sqrt {-a}\right ) \operatorname {erf}\left (x \sqrt {-a}\right )+c_{3} \left (x^{2} a -1\right )}{\operatorname {erf}\left (x \sqrt {-a}\right ) \sqrt {-a}\, \sqrt {\pi }\, x +{\mathrm e}^{x^{2} a}+c_{3} x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{a \,x^{2}} c_{1} a x -c_{1} \sqrt {\pi }\, \left (\left (-a \right )^{\frac {3}{2}} x^{2}+\sqrt {-a}\right ) \operatorname {erf}\left (\sqrt {-a}\, x \right )+a \,x^{2}-1}{\sqrt {\pi }\, \sqrt {-a}\, \operatorname {erf}\left (\sqrt {-a}\, x \right ) c_{1} x +{\mathrm e}^{a \,x^{2}} c_{1} +x} \] |
Problem 10332
| ODE |
\[ \boxed {y^{\prime }-y^{2}=x^{2} a^{2}+b x +c} \] |
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program solution |
\[ y = \frac {4 \left (-i a^{3}+\frac {1}{3} a^{2} c -\frac {1}{12} b^{2}\right ) \left (a^{2} x +\frac {b}{2}\right )^{2} \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +28 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {5}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )+\left (4 i a^{7} x^{2}+4 i a^{5} b x -4 a^{6}+i a^{3} b^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +12 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )+2 \left (a^{2} x +\frac {b}{2}\right ) \left (\left (a^{2} c -\frac {1}{4} b^{2}-i a^{3}\right ) \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +20 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )+i a^{3} \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +4 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )\right ) c_{3}}{4 \left (\left (a^{2} x +\frac {b}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +12 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )+\frac {\operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +4 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right ) c_{3}}{2}\right ) a^{4}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-48 \left (i a^{3}-\frac {1}{3} a^{2} c +\frac {1}{12} b^{2}\right ) \left (a^{2} x +\frac {b}{2}\right )^{2} c_{1} \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +28 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {5}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )+48 c_{1} a^{3} \left (i a^{4} x^{2}+i a^{2} b x +\frac {1}{4} i b^{2}-a^{3}\right ) \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +12 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )+24 \left (a^{2} x +\frac {b}{2}\right ) \left (\left (-i a^{3}+a^{2} c -\frac {1}{4} b^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +20 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )+i a^{3} \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +4 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )\right )}{48 \left (\left (a^{2} x +\frac {b}{2}\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +12 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )+\frac {\operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +4 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 a^{2} x +b \right )^{2}}{4 a^{3}}\right )}{2}\right ) a^{4}} \] |
Problem 10333
| ODE |
\[ \boxed {-a y^{2}+y^{\prime }=b \,x^{n}} \] |
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program solution |
\[ y = \frac {\operatorname {BesselJ}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{2+n}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}} c_{3} +\operatorname {BesselY}\left (\frac {n +3}{2+n}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{2+n}\right ) \sqrt {a b}\, x^{1+\frac {n}{2}}-\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{2+n}\right )-\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{2+n}\right ) c_{3}}{x a \left (\operatorname {BesselY}\left (\frac {1}{2+n}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{2+n}\right )+\operatorname {BesselJ}\left (\frac {1}{2+n}, \frac {2 \sqrt {a b}\, x^{1+\frac {n}{2}}}{2+n}\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\operatorname {BesselJ}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1} c_{1} +\operatorname {BesselY}\left (\frac {3+n}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1}-c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )}{x a \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )\right )} \] |
Problem 10334
| ODE |
\[ \boxed {y^{\prime }-y^{2}=a n \,x^{-1+n}-a^{2} x^{2 n}} \] |
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program solution |
\[ y = \frac {\left (-\frac {3 \,{\mathrm e}^{\frac {x \,x^{n} a}{1+n}} \left (2+n \right ) \left (\left (\frac {1}{3} n^{2}+n +\frac {2}{3}\right ) x^{-\frac {n}{2}}+a x \,x^{\frac {n}{2}} \left (n +\frac {4}{3}\right )\right ) \operatorname {WhittakerM}\left (\frac {2+n}{2+2 n}, \frac {3+2 n}{2+2 n}, -\frac {2 x \,x^{n} a}{1+n}\right )}{2}+{\mathrm e}^{\frac {x \,x^{n} a}{1+n}} \left (1+n \right ) \left (\left (-\frac {1}{2} n^{2}-\frac {3}{2} n -1\right ) x^{-\frac {n}{2}}+\left (\left (-\frac {n}{2}-1\right ) x^{\frac {n}{2}}+a x \,x^{\frac {3 n}{2}}\right ) x a \right ) \operatorname {WhittakerM}\left (-\frac {n}{2+2 n}, \frac {3+2 n}{2+2 n}, -\frac {2 x \,x^{n} a}{1+n}\right )+\left (\frac {3}{2}+n \right ) \left (2+n \right )^{2} x^{-\frac {n}{2}} {\mathrm e}^{\frac {2 x \,x^{n} a}{1+n}} \left (-\frac {2 x \,x^{n} a}{1+n}\right )^{\frac {4+3 n}{2+2 n}}+a c_{3} x^{2} x^{2 n}\right ) x^{-n}}{\left (-\frac {x^{-\frac {3 n}{2}} {\mathrm e}^{\frac {x \,x^{n} a}{1+n}} \left (2+n \right )^{2} \operatorname {WhittakerM}\left (\frac {2+n}{2+2 n}, \frac {3+2 n}{2+2 n}, -\frac {2 x \,x^{n} a}{1+n}\right )}{2}+{\mathrm e}^{\frac {x \,x^{n} a}{1+n}} \left (a x \,x^{-\frac {n}{2}}-\frac {\left (2+n \right ) x^{-\frac {3 n}{2}}}{2}\right ) \left (1+n \right ) \operatorname {WhittakerM}\left (-\frac {n}{2+2 n}, \frac {3+2 n}{2+2 n}, -\frac {2 x \,x^{n} a}{1+n}\right )+c_{3} x \right ) x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-3 \left (n +2\right ) c_{1} \left (\left (\frac {1}{3} n^{2}+n +\frac {2}{3}\right ) x^{-\frac {3 n}{2}}+a x \,x^{-\frac {n}{2}} \left (n +\frac {4}{3}\right )\right ) {\mathrm e}^{\frac {a x \,x^{n}}{n +1}} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x \,x^{n}}{n +1}\right )+2 c_{1} {\mathrm e}^{\frac {a x \,x^{n}}{n +1}} \left (n +1\right ) \left (\left (-\frac {1}{2} n^{2}-\frac {3}{2} n -1\right ) x^{-\frac {3 n}{2}}+x a \left (\left (-\frac {n}{2}-1\right ) x^{-\frac {n}{2}}+a x \,x^{\frac {n}{2}}\right )\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x \,x^{n}}{n +1}\right )+2 \left (n +2\right )^{2} c_{1} \left (n +\frac {3}{2}\right ) {\mathrm e}^{\frac {2 a x \,x^{n}}{n +1}} x^{-\frac {3 n}{2}} \left (-\frac {2 a x \,x^{n}}{n +1}\right )^{\frac {3 n +4}{2 n +2}}+2 x^{2} a \,x^{n}}{2 \left (-\frac {{\mathrm e}^{\frac {a x \,x^{n}}{n +1}} x^{-\frac {3 n}{2}} c_{1} \left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x \,x^{n}}{n +1}\right )}{2}+c_{1} \left (\left (-\frac {n}{2}-1\right ) x^{-\frac {3 n}{2}}+a x \,x^{-\frac {n}{2}}\right ) {\mathrm e}^{\frac {a x \,x^{n}}{n +1}} \left (n +1\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a x \,x^{n}}{n +1}\right )+x \right ) x} \] |
Problem 10335
| ODE |
\[ \boxed {-a y^{2}+y^{\prime }=b \,x^{2 n}+c \,x^{-1+n}} \] |
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program solution |
\[ y = \frac {\left (i \sqrt {a}\, \sqrt {b}\, c -b \left (2+n \right )\right ) c_{3} \operatorname {WhittakerM}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i \sqrt {a}\, c}{\sqrt {b}\, \left (2+2 n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{1+n}\right )+2 b \left (1+n \right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i \sqrt {a}\, c}{\sqrt {b}\, \left (2+2 n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{1+n}\right )-\left (2 i b^{\frac {3}{2}} x \,x^{n} \sqrt {a}+i \sqrt {a}\, \sqrt {b}\, c -b n \right ) \left (c_{3} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2+2 n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{1+n}\right )+\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2+2 n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{1+n}\right )\right )}{2 b x a \left (c_{3} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2+2 n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{1+n}\right )+\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2+2 n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{1+n}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (\left (\frac {n}{2}+1\right ) \sqrt {b}-\frac {i c \sqrt {a}}{2}\right ) \operatorname {WhittakerM}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i c \sqrt {a}}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right )-\sqrt {b}\, c_{1} \left (n +1\right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i c \sqrt {a}}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right )+\left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right )\right ) \left (-\frac {\sqrt {b}\, n}{2}+i \sqrt {a}\, \left (x \,x^{n} b +\frac {c}{2}\right )\right )}{\sqrt {b}\, \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right )\right ) a x} \] |
Problem 10336
| ODE |
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}=b \,x^{-n -2}} \] |
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program solution |
\[ y = \frac {\left (\left (-1-n +\sqrt {-4 a b +n^{2}+2 n +1}\right ) x^{-\frac {\sqrt {-4 a b +n^{2}+2 n +1}}{2}}-x^{\frac {\sqrt {-4 a b +n^{2}+2 n +1}}{2}} c_{3} \left (1+n +\sqrt {-4 a b +n^{2}+2 n +1}\right )\right ) x^{-n}}{2 x a \left (x^{\frac {\sqrt {-4 a b +n^{2}+2 n +1}}{2}} c_{3} +x^{-\frac {\sqrt {-4 a b +n^{2}+2 n +1}}{2}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x^{-n -1} \left (n +1-\tan \left (\frac {\sqrt {4 a b -n^{2}-2 n -1}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a b -n^{2}-2 n -1}\right )}{2 a} \] |
Problem 10337
| ODE |
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}=b \,x^{m}} \] |
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program solution |
\[ y = \frac {x^{\frac {m}{2}-\frac {n}{2}} \sqrt {a b}\, \left (\operatorname {BesselJ}\left (\frac {m +1}{m +n +2}, \frac {2 \sqrt {a b}\, x^{\frac {m}{2}+\frac {n}{2}+1}}{m +n +2}\right ) c_{3} +\operatorname {BesselY}\left (\frac {m +1}{m +n +2}, \frac {2 \sqrt {a b}\, x^{\frac {m}{2}+\frac {n}{2}+1}}{m +n +2}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-1-n}{m +n +2}, \frac {2 \sqrt {a b}\, x^{\frac {m}{2}+\frac {n}{2}+1}}{m +n +2}\right )+\operatorname {BesselJ}\left (\frac {-1-n}{m +n +2}, \frac {2 \sqrt {a b}\, x^{\frac {m}{2}+\frac {n}{2}+1}}{m +n +2}\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{-\frac {n}{2}+\frac {m}{2}} \sqrt {a b}\, \left (\operatorname {BesselY}\left (\frac {1+m}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1+m}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-n -1}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-n -1}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right )\right )} \] |
Problem 10338
| ODE |
\[ \boxed {y^{\prime }-y^{2}=k \left (x a +b \right )^{n} \left (c x +d \right )^{-n -4}} \] |
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program solution |
\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+k \left (x a +b \right )^{n} \left (c x +d \right )^{-n -4} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+k \left (x a +b \right )^{n} \left (c x +d \right )^{-n -4} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 10339
| ODE |
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}=b m \,x^{m -1}-a \,b^{2} x^{n +2 m}} \] |
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program solution |
\[ y = \frac {x^{-1-m -n} \left (-\frac {3 \left (m +2 n +2\right ) {\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \left (a b \left (m +\frac {4 n}{3}+\frac {4}{3}\right ) x^{1+n +\frac {m}{2}}+\frac {x^{-\frac {m}{2}} \left (m +2 n +2\right ) \left (1+m +n \right )}{3}\right ) \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2+2 m +2 n}, \frac {2 m +3 n +3}{2+2 m +2 n}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )}{2}+{\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \left (a^{2} b^{2} x^{\frac {3 m}{2}+2 n +2}-\frac {\left (m +2 n +2\right ) \left (a b \,x^{1+n +\frac {m}{2}}+x^{-\frac {m}{2}} \left (1+m +n \right )\right )}{2}\right ) \left (1+m +n \right ) \operatorname {WhittakerM}\left (-\frac {m}{2+2 m +2 n}, \frac {2 m +3 n +3}{2+2 m +2 n}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )+x^{-\frac {m}{2}} \left (m +2 n +2\right )^{2} \left (m +\frac {3 n}{2}+\frac {3}{2}\right ) {\mathrm e}^{\frac {2 a b \,x^{1+m +n}}{1+m +n}} \left (-\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )^{\frac {3 m +4 n +4}{2+2 m +2 n}}+a b c_{3} x^{2+2 m +2 n}\right )}{\left (-\frac {x^{-\frac {3 m}{2}} {\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \left (m +2 n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2+2 m +2 n}, \frac {2 m +3 n +3}{2+2 m +2 n}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )}{2}+{\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \left (x^{1-\frac {m}{2}+n} a b -\frac {x^{-\frac {3 m}{2}} \left (m +2 n +2\right )}{2}\right ) \left (1+m +n \right ) \operatorname {WhittakerM}\left (-\frac {m}{2+2 m +2 n}, \frac {2 m +3 n +3}{2+2 m +2 n}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )+x^{1+n} c_{3} \right ) a} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{-n -1} \left (-\frac {3 \left (a b \left (m +\frac {4 n}{3}+\frac {4}{3}\right ) x^{n +1-\frac {m}{2}}+\frac {x^{-\frac {3 m}{2}} \left (m +2 n +2\right ) \left (1+m +n \right )}{3}\right ) c_{1} \left (m +2 n +2\right ) {\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )}{2}+\left (a^{2} b^{2} x^{2 n +2+\frac {m}{2}}-\frac {\left (x^{n +1-\frac {m}{2}} a b +x^{-\frac {3 m}{2}} \left (1+m +n \right )\right ) \left (m +2 n +2\right )}{2}\right ) \left (1+m +n \right ) c_{1} {\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )+{\mathrm e}^{\frac {2 a b \,x^{1+m +n}}{1+m +n}} \left (m +\frac {3 n}{2}+\frac {3}{2}\right ) c_{1} \left (m +2 n +2\right )^{2} x^{-\frac {3 m}{2}} \left (-\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )^{\frac {3 m +4 n +4}{2 n +2 m +2}}+a b \,x^{m +2 n +2}\right )}{\left (-\frac {{\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} x^{-\frac {3 m}{2}} c_{1} \left (m +2 n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {m +2 n +2}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )}{2}+\left (1+m +n \right ) c_{1} \left (x^{n +1-\frac {m}{2}} a b -\frac {x^{-\frac {3 m}{2}} \left (m +2 n +2\right )}{2}\right ) {\mathrm e}^{\frac {a b \,x^{1+m +n}}{1+m +n}} \operatorname {WhittakerM}\left (-\frac {m}{2 n +2 m +2}, \frac {2 m +3 n +3}{2 n +2 m +2}, -\frac {2 a b \,x^{1+m +n}}{1+m +n}\right )+x^{n +1}\right ) a} \] |
Problem 10340
| ODE |
\[ \boxed {y^{\prime }-\left (a \,x^{2 n}+b \,x^{-1+n}\right ) y^{2}=c} \] |
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program solution |
\[ y = \frac {\left ({\mathrm e}^{\frac {i \left (-4 \sqrt {c}\, \sqrt {a}\, x^{1+n}+\pi \left (2+n \right )\right )}{4+4 n}} c \left (a^{4} x^{5+4 n}+4 a^{3} b \,x^{4+3 n}+6 a^{2} b^{2} x^{3+2 n}+4 a \,x^{2+n} b^{3}+x \,b^{4}\right ) \operatorname {hypergeom}\left (\left [\frac {\left (2+n \right ) \sqrt {a}+i \sqrt {c}\, b}{\sqrt {a}\, \left (2+2 n \right )}\right ], \left [\frac {2+n}{1+n}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}\right )+{\mathrm e}^{-\frac {i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {c}\, b +\sqrt {a}\, n}{\sqrt {a}\, \left (2+2 n \right )}\right ], \left [\frac {n}{1+n}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}\right ) c_{3} \left (a^{4} x^{4+4 n}+4 a^{3} b \,x^{3+3 n}+6 a^{2} x^{2+2 n} b^{2}+4 a \,x^{1+n} b^{3}+b^{4}\right )\right ) \left (2+n \right ) n \sqrt {c}}{\left (\left (2 \left (i a^{\frac {3}{2}} b n -a \sqrt {c}\, b^{2}\right ) x^{1+2 n}+\left (-a^{2} \sqrt {c}\, b +i a^{\frac {5}{2}} n \right ) x^{2+3 n}+b^{2} \left (-\sqrt {c}\, b +i \sqrt {a}\, n \right ) x^{n}\right ) {\mathrm e}^{-\frac {i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}} c_{3} \left (2+n \right ) \operatorname {hypergeom}\left (\left [\frac {\left (2+3 n \right ) \sqrt {a}+i \sqrt {c}\, b}{\sqrt {a}\, \left (2+2 n \right )}\right ], \left [\frac {1+2 n}{1+n}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}\right )+\left (\left (2 b \left (i c \left (2+n \right ) a^{\frac {3}{2}}-a b \,c^{\frac {3}{2}}\right ) x^{2+2 n}+\left (i c \left (2+n \right ) a^{\frac {5}{2}}-c^{\frac {3}{2}} a^{2} b \right ) x^{3+3 n}+x^{1+n} \left (-c^{\frac {3}{2}} b +i \sqrt {a}\, c \left (2+n \right )\right ) b^{2}\right ) {\mathrm e}^{\frac {i \left (-4 \sqrt {c}\, \sqrt {a}\, x^{1+n}+\pi \left (2+n \right )\right )}{4+4 n}} \operatorname {hypergeom}\left (\left [\frac {\left (4+3 n \right ) \sqrt {a}+i \sqrt {c}\, b}{\sqrt {a}\, \left (2+2 n \right )}\right ], \left [\frac {3+2 n}{1+n}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}\right )+\left (\left (\left (-2 i a^{\frac {3}{2}} b c +\sqrt {c}\, a^{2}\right ) x^{2+2 n}-i a^{\frac {5}{2}} x^{3+3 n} c +\left (\left (-i \sqrt {a}\, b c +2 a \sqrt {c}\right ) x^{1+n}+\sqrt {c}\, b \right ) b \right ) {\mathrm e}^{\frac {i \left (-4 \sqrt {c}\, \sqrt {a}\, x^{1+n}+\pi \left (2+n \right )\right )}{4+4 n}} \operatorname {hypergeom}\left (\left [\frac {\left (2+n \right ) \sqrt {a}+i \sqrt {c}\, b}{\sqrt {a}\, \left (2+2 n \right )}\right ], \left [\frac {2+n}{1+n}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}\right )-i {\mathrm e}^{-\frac {i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}} \left (\sqrt {a}\, b^{2} x^{n}+2 a^{\frac {3}{2}} x^{1+2 n} b +a^{\frac {5}{2}} x^{2+3 n}\right ) c_{3} \operatorname {hypergeom}\left (\left [\frac {i \sqrt {c}\, b +\sqrt {a}\, n}{\sqrt {a}\, \left (2+2 n \right )}\right ], \left [\frac {n}{1+n}\right ], \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{1+n}}{1+n}\right )\right ) \left (2+n \right )\right ) n \right ) \left (a \,x^{1+n}+b \right )^{2}} \] Verified OK.
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 10341
| ODE |
\[ \boxed {\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )=-a_{0} x -b_{0}} \] |
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program solution |
\[ y = \frac {\frac {\left (2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )\right ) c_{3} \operatorname {WhittakerM}\left (\frac {2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )}{2}-a_{2}^{\frac {3}{2}} \operatorname {WhittakerW}\left (\frac {2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right ) \sqrt {a_{0}}+i \left (c_{3} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+\operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right ) \left (\left (a_{0} x +\frac {b_{0}}{2}\right ) a_{2} +\frac {a_{0} b_{2}}{2}\right ) \sqrt {\lambda }}{\sqrt {a_{0}}\, \sqrt {a_{2}}\, \left (a_{2} x +b_{2} \right ) \lambda \left (c_{3} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+\operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (\frac {c_{1} \lambda \left (a_{0} b_{2} -a_{2} b_{0} \right ) \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )}{2}+\sqrt {-a_{2} \lambda a_{0}}\, a_{2} \left (c_{1} \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )+\lambda \operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )+\operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right ) \lambda \right )\right ) a_{0}}{\left (\frac {c_{1} \sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right ) \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )}{2}+a_{0} a_{2}^{2} \left (\operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right ) \lambda +c_{1} \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )-\lambda \operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )\right )\right ) \lambda } \] |
Problem 10342
| ODE |
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}=b} \] |
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program solution |
\[ y = \frac {\left (-1+\sqrt {-4 a b +1}\right ) x^{-\frac {\sqrt {-4 a b +1}}{2}}-c_{3} x^{\frac {\sqrt {-4 a b +1}}{2}} \left (1+\sqrt {-4 a b +1}\right )}{2 x a \left (c_{3} x^{\frac {\sqrt {-4 a b +1}}{2}}+x^{-\frac {\sqrt {-4 a b +1}}{2}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-1+\tan \left (\frac {\sqrt {4 a b -1}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a b -1}}{2 a x} \] |
Problem 10343
| ODE |
\[ \boxed {x^{2} y^{\prime }-x^{2} y^{2}=-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (1+b \right )} \] |
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program solution |
\[ y = \frac {-2 x^{2} a \left (-x^{2} a \right )^{b -\frac {1}{2}} {\mathrm e}^{x^{2} a}+\left (x^{2} a +b \right ) \left (-\Gamma \left (b +\frac {1}{2}\right )+\Gamma \left (b +\frac {1}{2}, -x^{2} a \right )-c_{3} \right )}{x \left (-\Gamma \left (b +\frac {1}{2}\right )+\Gamma \left (b +\frac {1}{2}, -x^{2} a \right )-c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-2 \left (-a \,x^{2}\right )^{b -\frac {1}{2}} {\mathrm e}^{a \,x^{2}} c_{1} a \,x^{2}+\left (-c_{1} \Gamma \left (b +\frac {1}{2}\right )+c_{1} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )-1\right ) \left (a \,x^{2}+b \right )}{x \left (-c_{1} \Gamma \left (b +\frac {1}{2}\right )+c_{1} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )-1\right )} \] |
Problem 10344
| ODE |
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}=b \,x^{n}+c} \] |
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program solution |
\[ y = \frac {2 \sqrt {a b}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 c a +1}}{n}+1, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 c a +1}}{n}+1, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-\left (\operatorname {BesselY}\left (\frac {\sqrt {-4 c a +1}}{n}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right )+\operatorname {BesselJ}\left (\frac {\sqrt {-4 c a +1}}{n}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} \right ) \left (\sqrt {-4 c a +1}+1\right )}{2 x a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 c a +1}}{n}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right )+\operatorname {BesselJ}\left (\frac {\sqrt {-4 c a +1}}{n}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 \sqrt {a b}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{n}+1, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{n}+1, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-\left (\sqrt {-4 a c +1}+1\right ) \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{n}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{n}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right )\right )}{2 x a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{n}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{n}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}}}{n}\right )\right )} \] |
Problem 10345
| ODE |
\[ \boxed {x^{2} y^{\prime }-x^{2} y^{2}=a \,x^{2 m} \left (b \,x^{m}+c \right )^{n}-\frac {n^{2}}{4}+\frac {1}{4}} \] |
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program solution |
\[ y = -\frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {4 x^{2 m} \left (b \,x^{m}+c \right )^{n} \textit {\_Y} \left (x \right ) a +4 \textit {\_Y}^{\prime \prime }\left (x \right ) x^{2}-\textit {\_Y} \left (x \right ) n^{2}+\textit {\_Y} \left (x \right )}{4 x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{\frac {4 x^{2 m} \left (b \,x^{m}+c \right )^{n} \textit {\_Y} \left (x \right ) a +4 \textit {\_Y}^{\prime \prime }\left (x \right ) x^{2}-\textit {\_Y} \left (x \right ) n^{2}+\textit {\_Y} \left (x \right )}{4 x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 10346
| ODE |
\[ \boxed {\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )=-a_{0}} \] |
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program solution |
\[ y = -\frac {4 \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1+\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right ) c_{4} \left (-1+\sqrt {-4 a_{0} \lambda +1}\right ) \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}}-\left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1-\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right ) c_{3} \left (1+\sqrt {-4 a_{0} \lambda +1}\right )\right ) \left (\left (x +\frac {b_{2}}{2}\right ) \sqrt {b_{2}^{2}-4 a_{2}}+x^{2}+b_{2} x +\frac {b_{2}^{2}}{2}-a_{2} \right )}{\left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{2} \left (-\sqrt {b_{2}^{2}-4 a_{2}}+2 x +b_{2} \right ) \lambda \left (c_{4} \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {3}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1+\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right )+c_{3} \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {3}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1-\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right )\right )} \] Verified OK.
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 10347
| ODE |
\[ \boxed {x^{4} y^{\prime }+x^{4} y^{2}=-a^{2}} \] |
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program solution |
\[ y = \frac {\left (-a c_{3} +x \right ) \cos \left (\frac {a}{x}\right )+\sin \left (\frac {a}{x}\right ) \left (c_{3} x +a \right )}{\left (c_{3} \sin \left (\frac {a}{x}\right )+\cos \left (\frac {a}{x}\right )\right ) x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-a \tan \left (\frac {a \left (c_{1} x -1\right )}{x}\right )+x}{x^{2}} \] |
Problem 10348
| ODE |
\[ \boxed {a \,x^{2} \left (x -1\right )^{2} \left (y^{\prime }+\lambda y^{2}\right )=-b \,x^{2}-c x -s} \] |
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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 10349
| ODE |
\[ \boxed {\left (x^{2} a +b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )=-A} \] |
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program solution |
\[ y = \frac {2 a \left (\left (i \sqrt {4 c a -b^{2}}\, \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}\, a -2 \left (x a +\frac {b}{2}\right ) \sqrt {-4 c a +b^{2}}\right ) {\left (\frac {-b +i \sqrt {4 c a -b^{2}}-2 x a}{i \sqrt {4 c a -b^{2}}+2 x a +b}\right )}^{-\frac {a \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}}-{\left (\frac {-b +i \sqrt {4 c a -b^{2}}-2 x a}{i \sqrt {4 c a -b^{2}}+2 x a +b}\right )}^{\frac {a \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \left (i \sqrt {4 c a -b^{2}}\, \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}\, a +2 \left (x a +\frac {b}{2}\right ) \sqrt {-4 c a +b^{2}}\right ) c_{3} \right )}{\sqrt {-4 c a +b^{2}}\, \left ({\left (\frac {-b +i \sqrt {4 c a -b^{2}}-2 x a}{i \sqrt {4 c a -b^{2}}+2 x a +b}\right )}^{-\frac {a \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}}+{\left (\frac {-b +i \sqrt {4 c a -b^{2}}-2 x a}{i \sqrt {4 c a -b^{2}}+2 x a +b}\right )}^{\frac {a \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} c_{3} \right ) \left (-b +i \sqrt {4 c a -b^{2}}-2 x a \right ) \left (i \sqrt {4 c a -b^{2}}+2 x a +b \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 \left (c_{1} \left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}\, a \sqrt {4 a c -b^{2}}-2 \sqrt {-4 a c +b^{2}}\, \left (\frac {b}{2}+a x \right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-\left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}\, a \sqrt {4 a c -b^{2}}+2 \sqrt {-4 a c +b^{2}}\, \left (\frac {b}{2}+a x \right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right ) a}{\sqrt {-4 a c +b^{2}}\, \left (i \sqrt {4 a c -b^{2}}+2 a x +b \right ) \left (-b +i \sqrt {4 a c -b^{2}}-2 a x \right ) \left (c_{1} {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )} \] |
Problem 10350
| ODE |
\[ \boxed {x^{1+n} y^{\prime }-x^{2 n} y^{2} a=c \,x^{m}+d} \] |
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program solution |
\[ y = -\frac {x^{-n} \left (-2 \sqrt {c a}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}+1, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}+1, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right )\right ) x^{\frac {m}{2}}+\left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right )\right ) \left (\sqrt {-4 a d +n^{2}}+n \right )\right )}{2 a \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}}}{m}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x^{-n} \left (-2 \sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}+1, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}+1, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right ) x^{\frac {m}{2}}+\left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right ) \left (\sqrt {-4 a d +n^{2}}+n \right )\right )}{2 a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +n^{2}}}{m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}}}{m}\right )\right )} \] |
Problem 10351
| ODE |
\[ \boxed {\left (x^{n} a +b \right ) y^{\prime }-y^{2} b=a \,x^{-2+n}} \] |
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program solution |
\[ y = \frac {a \left (\frac {x^{n} a +b}{b}\right )^{\frac {2}{n}} n \left (a^{2} x^{3 n}+2 x^{2 n} a b +b^{2} x^{n}\right ) \operatorname {hypergeom}\left (\left [2, \frac {1+n}{n}\right ], \left [\frac {2 n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right )-\left (-1+n \right ) b \left (a \left (\frac {x^{n} a +b}{b}\right )^{\frac {2}{n}} \left (a \,x^{2 n}+b \,x^{n}\right ) \operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {-1+n}{n}\right ], -\frac {a \,x^{n}}{b}\right )+b c_{3} \left (a \,x^{1+n}+b x \right )\right )}{b^{2} \left (-1+n \right ) x \left (c_{3} x +\operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {-1+n}{n}\right ], -\frac {a \,x^{n}}{b}\right ) \left (\frac {x^{n} a +b}{b}\right )^{\frac {2}{n}}\right ) \left (x^{n} a +b \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (\frac {a \,x^{n}+b}{b}\right )^{\frac {2}{n}} \left (a n c_{1} \left (a^{2} x^{3 n}+2 a b \,x^{2 n}+b^{2} x^{n}\right ) \operatorname {hypergeom}\left (\left [2, \frac {n +1}{n}\right ], \left [\frac {2 n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right )-\left (n -1\right ) b \left (a c_{1} \left (a \,x^{2 n}+b \,x^{n}\right ) \operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right )+\left (\frac {a \,x^{n}+b}{b}\right )^{-\frac {2}{n}} b \left (x^{n +1} a +b x \right )\right )\right )}{b^{2} \left (n -1\right ) x \left (a \,x^{n}+b \right ) \left (x +\operatorname {hypergeom}\left (\left [1, \frac {1}{n}\right ], \left [\frac {n -1}{n}\right ], -\frac {a \,x^{n}}{b}\right ) c_{1} \left (\frac {a \,x^{n}+b}{b}\right )^{\frac {2}{n}}\right )} \] |
Problem 10352
| ODE |
\[ \boxed {\left (x^{n} a +b \,x^{m}+c \right ) \left (y^{\prime }-y^{2}\right )=-a n \left (-1+n \right ) x^{-2+n}-b m \left (m -1\right ) x^{m -2}} \] |
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program solution |
\[ y = \frac {-\frac {c_{3}}{x^{n} a +b \,x^{m}+c}-\frac {\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{3} +1\right ) \left (x^{n} n a +b \,x^{m} m \right )}{x}}{\left (\left (\int \frac {1}{\left (x^{n} a +b \,x^{m}+c \right )^{2}}d x \right ) c_{3} +1\right ) \left (x^{n} a +b \,x^{m}+c \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\left (a n \,x^{n}+b m \,x^{m}\right ) \left (a \,x^{n}+b \,x^{m}+c \right ) \left (\int \frac {1}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2}}d x \right )-x^{2 m} c_{1} b^{2} m -b \left (a \left (n +m \right ) x^{n}+c m \right ) c_{1} x^{m}-x^{2 n} c_{1} a^{2} n -x^{n} c_{1} a c n -x}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2} x \left (c_{1} +\int \frac {1}{\left (a \,x^{n}+b \,x^{m}+c \right )^{2}}d x \right )} \] |
Problem 10353
| ODE |
\[ \boxed {-a y^{2}+y^{\prime }-y b=c x +k} \] |
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program solution |
\[ y = \frac {-\operatorname {AiryAi}\left (-\frac {\left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right ) \left (c a \right )^{\frac {1}{3}}}{a c}\right ) c_{3} b -b \operatorname {AiryBi}\left (-\frac {\left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right ) \left (c a \right )^{\frac {1}{3}}}{a c}\right )+2 \left (c a \right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\frac {\left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right ) \left (c a \right )^{\frac {1}{3}}}{a c}\right ) c_{3} +\operatorname {AiryBi}\left (1, -\frac {\left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right ) \left (c a \right )^{\frac {1}{3}}}{a c}\right )\right )}{2 a \left (\operatorname {AiryAi}\left (-\frac {\left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right ) \left (c a \right )^{\frac {1}{3}}}{a c}\right ) c_{3} +\operatorname {AiryBi}\left (-\frac {\left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right ) \left (c a \right )^{\frac {1}{3}}}{a c}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 \sqrt {a}\, \left (\frac {c}{\sqrt {a}}\right )^{\frac {1}{3}} \left (\operatorname {AiryAi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right ) c_{1} +\operatorname {AiryBi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right )-b \left (c_{1} \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right )}{2 a \left (c_{1} \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{\frac {2}{3}} a}\right )\right )} \] |
Problem 10354
| ODE |
\[ \boxed {y^{\prime }-y^{2}-a \,x^{n} y=a \,x^{-1+n}} \] |
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program solution |
\[ y = -\frac {\left ({\mathrm e}^{\frac {x^{n} a x}{2+2 n}} \left (1+n \right ) \left (a x \,x^{\frac {n}{2}}+x^{-\frac {n}{2}} n^{2}\right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2+2 n}, \frac {1+2 n}{2+2 n}, -\frac {x \,x^{n} a}{1+n}\right )+{\mathrm e}^{\frac {x^{n} a x}{2+2 n}} n \left (x^{-\frac {n}{2}} n^{2}+a x \,x^{\frac {n}{2}} \left (1+n \right )\right ) \operatorname {WhittakerM}\left (\frac {n}{2+2 n}, \frac {1+2 n}{2+2 n}, -\frac {x \,x^{n} a}{1+n}\right )-2 x^{-\frac {n}{2}} n^{2} {\mathrm e}^{\frac {x \,x^{n} a}{1+n}} \left (\frac {1}{2}+n \right ) \left (-\frac {x \,x^{n} a}{1+n}\right )^{\frac {2+3 n}{2+2 n}}+c_{3} x^{2} x^{n}\right ) x^{-n}}{\left ({\mathrm e}^{\frac {x^{n} a x}{2+2 n}} \left (1+n \right ) \left (x^{-\frac {n}{2}} a x -n \,x^{-\frac {3 n}{2}}\right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2+2 n}, \frac {1+2 n}{2+2 n}, -\frac {x \,x^{n} a}{1+n}\right )-\operatorname {WhittakerM}\left (\frac {n}{2+2 n}, \frac {1+2 n}{2+2 n}, -\frac {x \,x^{n} a}{1+n}\right ) x^{-\frac {3 n}{2}} {\mathrm e}^{\frac {x^{n} a x}{2+2 n}} n^{2}+c_{3} x^{2}\right ) x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-{\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}} \left (n +1\right )^{2} \left (x^{n} a x -n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )-2 n \left (-\frac {\left (n +1\right ) n \,{\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )}{2}+\left (n +\frac {1}{2}\right ) x \,x^{n} a \left (c_{1} x -{\mathrm e}^{\frac {a x \,x^{n}}{n +1}}\right )\right )}{x \left ({\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}} \left (n +1\right )^{2} \left (x^{n} a x -n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )+2 n \left (-\frac {\left (n +1\right ) n \,{\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (-\frac {a x \,x^{n}}{n +1}\right )^{-\frac {n}{2 n +2}} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a x \,x^{n}}{n +1}\right )}{2}+a \,x^{2} c_{1} x^{n} \left (n +\frac {1}{2}\right )\right )\right )} \] |
Problem 10355
| ODE |
\[ \boxed {y^{\prime }-y^{2}-a \,x^{n} y=b \,x^{-1+n}} \] |
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program solution |
\[ y = \frac {-a c_{3} \left (1+n \right ) \left (a -b \right ) \operatorname {KummerM}\left (\frac {2+n -\frac {b}{a}}{1+n}, \frac {2+n}{1+n}, \frac {x \,x^{n} a}{1+n}\right )+\left (\left (a -b \right ) \operatorname {KummerU}\left (\frac {2+n -\frac {b}{a}}{1+n}, \frac {2+n}{1+n}, \frac {x \,x^{n} a}{1+n}\right )-a \left (\operatorname {KummerU}\left (\frac {a -b}{a \left (1+n \right )}, \frac {2+n}{1+n}, \frac {x \,x^{n} a}{1+n}\right )+\operatorname {KummerM}\left (\frac {a -b}{a \left (1+n \right )}, \frac {2+n}{1+n}, \frac {x \,x^{n} a}{1+n}\right ) c_{3} \right ) \left (1+n \right )\right ) b}{a^{2} \left (1+n \right ) x \left (\operatorname {KummerU}\left (\frac {a -b}{a \left (1+n \right )}, \frac {2+n}{1+n}, \frac {x \,x^{n} a}{1+n}\right )+\operatorname {KummerM}\left (\frac {a -b}{a \left (1+n \right )}, \frac {2+n}{1+n}, \frac {x \,x^{n} a}{1+n}\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-a \left (n +1\right ) \left (a -b \right ) \operatorname {KummerM}\left (\frac {a \left (n +2\right )-b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )+\left (\left (a -b \right ) c_{1} \operatorname {KummerU}\left (\frac {2+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )-a \left (\operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right ) c_{1} +\operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )\right ) \left (n +1\right )\right ) b}{a^{2} \left (n +1\right ) x \left (\operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right ) c_{1} +\operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )\right )} \] |
Problem 10356
| ODE |
\[ \boxed {y^{\prime }-y^{2}-\left (\alpha x +\beta \right ) y=x^{2} a +b x +c} \] |
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program solution |
\[ \text {Expression too large to display} \] Verified OK.
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 10357
| ODE |
\[ \boxed {y^{\prime }-y^{2}-a \,x^{n} y=-b a \,x^{n}-b^{2}} \] |
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program solution |
\[ y = \frac {\left (\int {\mathrm e}^{\frac {a \,x^{1+n}+2 b x \left (1+n \right )}{1+n}}d x -c_{3} \right ) b -{\mathrm e}^{\frac {a \,x^{1+n}+2 b x \left (1+n \right )}{1+n}}}{\int {\mathrm e}^{\frac {\left (x^{n} a +2 b \left (1+n \right )\right ) x}{1+n}}d x -c_{3}} \] Verified OK.
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Maple solution | \[ \frac {\left (b -y \left (x \right )\right ) \left (\int _{}^{x}{\mathrm e}^{\frac {\left (\textit {\_a}^{n} a +2 b \left (n +1\right )\right ) \textit {\_a}}{n +1}}d \textit {\_a} \right )+c_{1} b -c_{1} y \left (x \right )-{\mathrm e}^{\frac {\left (a \,x^{n}+2 b \left (n +1\right )\right ) x}{n +1}}}{b -y \left (x \right )} = 0 \] |
Problem 10358
| ODE |
\[ \boxed {y^{\prime }+\left (1+n \right ) x^{n} y^{2}=a \,x^{1+m +n}-a \,x^{m}} \] |
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program solution |
\[ y = \frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-a \textit {\_Y} \left (x \right ) \left (1+n \right ) x^{m +2 n +2}+a \textit {\_Y} \left (x \right ) \left (1+n \right ) x^{1+m +n}-n \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) x}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{\left (1+n \right ) \operatorname {DESol}\left (\left \{\frac {-a \textit {\_Y} \left (x \right ) \left (1+n \right ) x^{m +2 n +2}+a \textit {\_Y} \left (x \right ) \left (1+n \right ) x^{1+m +n}-n \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) x}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 10359
| ODE |
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}-b \,x^{m} y=x^{m} b c -a \,c^{2} x^{n}} \] |
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program solution |
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-x^{2 n} \textit {\_Y} \left (x \right ) a^{2} c^{2} x +x^{m +n} \textit {\_Y} \left (x \right ) a b c x -x^{m +1} \textit {\_Y}^{\prime }\left (x \right ) b +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x^{1+2 n} a^{2} c^{2} \textit {\_Y} \left (x \right )+a b c \,x^{1+m +n} \textit {\_Y} \left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right ) x -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \frac {a \left (c +y \left (x \right )\right ) \left (\int _{}^{x}{\mathrm e}^{-\frac {2 \left (-\frac {b \left (n +1\right ) \textit {\_a}^{m}}{2}+a \,\textit {\_a}^{n} c \left (1+m \right )\right ) \textit {\_a}}{\left (1+m \right ) \left (n +1\right )}} \textit {\_a}^{n}d \textit {\_a} \right )+c_{1} y \left (x \right )+c_{1} c +{\mathrm e}^{-\frac {2 \left (-\frac {b \left (n +1\right ) x^{m}}{2}+a \,x^{n} c \left (1+m \right )\right ) x}{\left (1+m \right ) \left (n +1\right )}}}{c +y \left (x \right )} = 0 \] |
Problem 10360
| ODE |
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}+a \,x^{n} \left (b \,x^{m}+c \right ) y=b m \,x^{m -1}} \] |
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program solution |
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {a b m \,x^{m +n} \textit {\_Y} \left (x \right )+x^{1+m +n} \textit {\_Y}^{\prime }\left (x \right ) a b +x^{1+n} \textit {\_Y}^{\prime }\left (x \right ) a c +\textit {\_Y}^{\prime \prime }\left (x \right ) x -n \textit {\_Y}^{\prime }\left (x \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 10361
| ODE |
\[ \boxed {y^{\prime }+a n \,x^{-1+n} y^{2}-c \,x^{m} \left (x^{n} a +b \right ) y=-c \,x^{m}} \] |
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program solution |
\[ y = \frac {a n \left (x^{n} a +b \right ) \left (\int \frac {x^{n} {\mathrm e}^{\frac {\left (a \left (m +1\right ) x^{n}+b \left (1+m +n \right )\right ) c \,x^{m} x}{\left (m +1\right ) \left (1+m +n \right )}}}{x \left (x^{n} a +b \right )^{2}}d x \right )+a^{2} n c_{3} x^{n}+a n c_{3} b +{\mathrm e}^{\frac {\left (a \left (m +1\right ) x^{n}+b \left (1+m +n \right )\right ) c \,x^{m} x}{\left (m +1\right ) \left (1+m +n \right )}}}{a n \left (x^{n} a +b \right )^{2} \left (\int \frac {x^{n} {\mathrm e}^{\frac {\left (a \left (m +1\right ) x^{n}+b \left (1+m +n \right )\right ) c \,x^{m} x}{\left (m +1\right ) \left (1+m +n \right )}}}{x \left (x^{n} a +b \right )^{2}}d x +c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {a n \left (a \,x^{n}+b \right ) \left (\int \frac {x^{n -1} {\mathrm e}^{\frac {c \left (a \left (1+m \right ) x^{1+m +n}+b \,x^{1+m} \left (1+m +n \right )\right )}{\left (1+m \right ) \left (1+m +n \right )}}}{\left (a \,x^{n}+b \right )^{2}}d x \right )-x^{n} c_{1} a -c_{1} b +{\mathrm e}^{\frac {c \left (a \left (1+m \right ) x^{1+m +n}+b \,x^{1+m} \left (1+m +n \right )\right )}{\left (1+m \right ) \left (1+m +n \right )}}}{\left (a \left (\int \frac {x^{n -1} {\mathrm e}^{\frac {x \left (a \left (1+m \right ) x^{n}+b \left (1+m +n \right )\right ) c \,x^{m}}{\left (1+m \right ) \left (1+m +n \right )}}}{\left (a \,x^{n}+b \right )^{2}}d x \right ) n -c_{1} \right ) \left (a^{2} x^{2 n}+2 x^{n} a b +b^{2}\right )} \] |
Problem 10362
| ODE |
\[ \boxed {y^{\prime }-a \,x^{n} y^{2}-b \,x^{m} y=c k \,x^{k -1}-b c \,x^{k +m}-a \,c^{2} x^{n +2 k}} \] |
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program solution |
\[ y = -\frac {\left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {-x^{n +2 k} x^{1+n} \textit {\_Y} \left (x \right ) a^{2} c^{2}+a c \textit {\_Y} \left (x \right ) \left (k \,x^{k -1}-x^{k +m} b \right ) x^{1+n}+\textit {\_Y}^{\prime \prime }\left (x \right ) x -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x^{-n}}{a \operatorname {DESol}\left (\left \{\frac {-x^{2 n +2 k +1} \textit {\_Y} \left (x \right ) a^{2} c^{2}-x^{k +1+m +n} \textit {\_Y} \left (x \right ) a b c +\textit {\_Y}^{\prime \prime }\left (x \right ) x +x^{k +n} \textit {\_Y} \left (x \right ) a c k -\textit {\_Y}^{\prime }\left (x \right ) \left (b \,x^{m +1}+n \right )}{x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 10363
| ODE |
\[ \boxed {y^{\prime } x -a y^{2}-y b=c \,x^{2 b}} \] |
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program solution |
\[ y = \frac {\left (-c_{3} \cos \left (\frac {x^{b} \sqrt {c a}}{b}\right )+\sin \left (\frac {x^{b} \sqrt {c a}}{b}\right )\right ) x^{b} \sqrt {c a}}{\left (c_{3} \sin \left (\frac {x^{b} \sqrt {c a}}{b}\right )+\cos \left (\frac {x^{b} \sqrt {c a}}{b}\right )\right ) a} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\tan \left (\frac {x^{b} \sqrt {a}\, \sqrt {c}-c_{1} b}{b}\right ) \sqrt {c}\, x^{b}}{\sqrt {a}} \] |
Problem 10364
| ODE |
\[ \boxed {y^{\prime } x -a y^{2}-y b=c \,x^{n}} \] |
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program solution |
\[ y = \frac {\sqrt {c a}\, \left (\operatorname {BesselJ}\left (\frac {b +n}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {b +n}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-b \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )+\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) c_{3} \right )}{a \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )+\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {b +n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-b \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \] |
Problem 10365
| ODE |
\[ \boxed {y^{\prime } x -a y^{2}-\left (n +b \,x^{n}\right ) y=c \,x^{2 n}} \] |
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program solution |
\[ y = -\frac {\left (\left (\sqrt {\frac {-4 c a +b^{2}}{n^{2}}}\, n c_{3} +b \right ) \cosh \left (\frac {x^{n} \sqrt {\frac {-4 c a +b^{2}}{n^{2}}}}{2}\right )+\sinh \left (\frac {x^{n} \sqrt {\frac {-4 c a +b^{2}}{n^{2}}}}{2}\right ) \left (\sqrt {\frac {-4 c a +b^{2}}{n^{2}}}\, n +b c_{3} \right )\right ) x^{n}}{2 a \left (c_{3} \sinh \left (\frac {x^{n} \sqrt {\frac {-4 c a +b^{2}}{n^{2}}}}{2}\right )+\cosh \left (\frac {x^{n} \sqrt {\frac {-4 c a +b^{2}}{n^{2}}}}{2}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x^{n} \left (b^{2}-\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\sqrt {4 a \,b^{2} c -b^{4}}\, \left (b \,x^{n}+c_{1} n \right )}{2 b^{2} n}\right )\right )}{2 a b} \] |
Problem 10366
| ODE |
\[ \boxed {y^{\prime } x -y^{2} x -a y=b \,x^{n}} \] |
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program solution |
\[ y = \frac {\sqrt {b}\, x^{-\frac {1}{2}+\frac {n}{2}} \left (\operatorname {BesselJ}\left (\frac {-a +n}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {1}{2}+\frac {n}{2}}}{1+n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {-a +n}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {1}{2}+\frac {n}{2}}}{1+n}\right )\right )}{\operatorname {BesselY}\left (\frac {-a -1}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {1}{2}+\frac {n}{2}}}{1+n}\right )+\operatorname {BesselJ}\left (\frac {-a -1}{1+n}, \frac {2 \sqrt {b}\, x^{\frac {1}{2}+\frac {n}{2}}}{1+n}\right ) c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{\frac {n}{2}-\frac {1}{2}} \sqrt {b}\, \left (\operatorname {BesselY}\left (\frac {-a +n}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-a +n}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )\right )}{\operatorname {BesselY}\left (\frac {-a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )} \] |
Problem 10367
| ODE |
\[ \boxed {y^{\prime } x +a_{3} x y^{2}+a_{2} y=-a_{1} x -a_{0}} \] |
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program solution |
\[ y = \frac {\frac {\left (-\left (\frac {1}{2} a_{2}^{2}-a_{2} \right ) a_{1}^{\frac {3}{2}}+i a_{1} \sqrt {a_{3}}\, a_{0} -\frac {a_{0}^{2} a_{3} \sqrt {a_{1}}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}+\frac {c_{3} \left (i a_{1} \sqrt {a_{3}}\, a_{0} +a_{2} a_{1}^{\frac {3}{2}}\right ) \operatorname {KummerM}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}-\left (\frac {a_{2} a_{1}^{\frac {3}{2}}}{2}+i \sqrt {a_{3}}\, \left (a_{1} x +\frac {a_{0}}{2}\right ) a_{1} \right ) \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{a_{1}^{\frac {3}{2}} x a_{3} \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {4 a_{1} \left (a_{1}^{3} a_{3} \left (a_{3} a_{0} -a_{2} \sqrt {-a_{1} a_{3}}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-\frac {c_{1} \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )}{4}+a_{1}^{3} a_{3} \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{3} a_{0} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_{1} \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right )}{2}\right )}{4 a_{1}^{3} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-c_{1} \sqrt {-a_{1} a_{3}}\, \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+2 a_{1} \left (-2 a_{1}^{2} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_{1} \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{3} a_{0} \right )\right )} \] |
Problem 10368
| ODE |
\[ \boxed {y^{\prime } x -a \,x^{n} y^{2}-y b=c \,x^{-n}} \] |
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program solution |
\[ y = \frac {\left (\left (-b -n +\sqrt {-4 c a +b^{2}+2 b n +n^{2}}\right ) x^{-\frac {\sqrt {-4 c a +b^{2}+2 b n +n^{2}}}{2}}-x^{\frac {\sqrt {-4 c a +b^{2}+2 b n +n^{2}}}{2}} c_{3} \left (b +n +\sqrt {-4 c a +b^{2}+2 b n +n^{2}}\right )\right ) x^{-n}}{2 a \left (x^{\frac {\sqrt {-4 c a +b^{2}+2 b n +n^{2}}}{2}} c_{3} +x^{-\frac {\sqrt {-4 c a +b^{2}+2 b n +n^{2}}}{2}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{-n} \left (\tan \left (\frac {\sqrt {4 a c -b^{2}-2 b n -n^{2}}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a c -b^{2}-2 b n -n^{2}}-b -n \right )}{2 a} \] |
Problem 10369
| ODE |
\[ \boxed {y^{\prime } x -a \,x^{n} y^{2}-m y=-a \,b^{2} x^{n +2 m}} \] |
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program solution |
\[ y = -\frac {b \,x^{m} \left (c_{3} \cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )+\sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )\right )}{c_{3} \sinh \left (\frac {a b \,x^{m +n}}{m +n}\right )+\cosh \left (\frac {a b \,x^{m +n}}{m +n}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = i \tan \left (\frac {c_{1} \left (n +m \right )+i a b \,x^{n +m}}{n +m}\right ) b \,x^{m} \] |
Problem 10370
| ODE |
\[ \boxed {y^{\prime } x -x^{2 n} y^{2}-\left (m -n \right ) y=x^{2 m}} \] |
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program solution |
\[ y = \frac {x^{m -n} \left (-c_{3} \cos \left (\frac {x^{m +n}}{m +n}\right )+\sin \left (\frac {x^{m +n}}{m +n}\right )\right )}{c_{3} \sin \left (\frac {x^{m +n}}{m +n}\right )+\cos \left (\frac {x^{m +n}}{m +n}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tan \left (\frac {x^{n +m}+\left (-n -m \right ) c_{1}}{n +m}\right ) x^{-n +m} \] |
Problem 10371
| ODE |
\[ \boxed {y^{\prime } x -a \,x^{n} y^{2}-y b=c \,x^{m}} \] |
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program solution |
\[ y = \frac {x^{\frac {m}{2}-\frac {n}{2}} \sqrt {c a}\, \left (\operatorname {BesselJ}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {-b +m}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right )+\operatorname {BesselJ}\left (\frac {-b -n}{m +n}, \frac {2 \sqrt {c a}\, x^{\frac {m}{2}+\frac {n}{2}}}{m +n}\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{-\frac {n}{2}+\frac {m}{2}} \sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {-b +m}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{n +m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-b +m}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{n +m}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-b -n}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{n +m}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-b -n}{n +m}, \frac {2 \sqrt {a c}\, x^{\frac {m}{2}+\frac {n}{2}}}{n +m}\right )\right )} \] |
Problem 10372
| ODE |
\[ \boxed {y^{\prime } x -x^{2 n} y^{2} a -\left (b \,x^{n}-n \right ) y=c} \] |
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program solution |
\[ y = -\frac {x^{-n} \left (\left (\sqrt {\frac {-4 c a +b^{2}}{n^{2}}}\, n c_{3} +b \right ) \cosh \left (\frac {x^{n} \sqrt {\frac {-4 c a +b^{2}}{n^{2}}}}{2}\right )+\sinh \left (\frac {x^{n} \sqrt {\frac {-4 c a +b^{2}}{n^{2}}}}{2}\right ) \left (\sqrt {\frac {-4 c a +b^{2}}{n^{2}}}\, n +b c_{3} \right )\right )}{2 a \left (c_{3} \sinh \left (\frac {x^{n} \sqrt {\frac {-4 c a +b^{2}}{n^{2}}}}{2}\right )+\cosh \left (\frac {x^{n} \sqrt {\frac {-4 c a +b^{2}}{n^{2}}}}{2}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\sqrt {4 a \,b^{2} c -b^{4}}\, \left (b \,x^{n}+c_{1} n \right )}{2 b^{2} n}\right )-b^{2}\right ) x^{-n}}{2 b a} \] |
Problem 10373
| ODE |
\[ \boxed {y^{\prime } x -a \,x^{m +2 n} y^{2}-\left (b \,x^{m +n}-n \right ) y=c \,x^{m}} \] |
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program solution |
\[ y = -\frac {x^{-n} \left (\left (c_{3} \left (m +n \right ) \sqrt {\frac {-4 c a +b^{2}}{\left (m +n \right )^{2}}}+b \right ) \cosh \left (\frac {x^{m +n} \sqrt {\frac {-4 c a +b^{2}}{\left (m +n \right )^{2}}}}{2}\right )+\left (\left (m +n \right ) \sqrt {\frac {-4 c a +b^{2}}{\left (m +n \right )^{2}}}+b c_{3} \right ) \sinh \left (\frac {x^{m +n} \sqrt {\frac {-4 c a +b^{2}}{\left (m +n \right )^{2}}}}{2}\right )\right )}{2 a \left (c_{3} \sinh \left (\frac {x^{m +n} \sqrt {\frac {-4 c a +b^{2}}{\left (m +n \right )^{2}}}}{2}\right )+\cosh \left (\frac {x^{m +n} \sqrt {\frac {-4 c a +b^{2}}{\left (m +n \right )^{2}}}}{2}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{-n} \left (\sqrt {4 a \,b^{2} c -b^{4}}\, \tan \left (\frac {\left (x^{n +m} b +c_{1} \left (n +m \right )\right ) \sqrt {4 a \,b^{2} c -b^{4}}}{2 b^{2} \left (n +m \right )}\right )-b^{2}\right )}{2 a b} \] |
Problem 10374
| ODE |
\[ \boxed {\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (a_{1} x +b_{1} \right ) y=-a_{0} x -b_{0}} \] |
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program solution |
\[ \text {Expression too large to display} \] Verified OK.
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 10375
| ODE |
\[ \boxed {\left (x a +c \right ) y^{\prime }-\alpha \left (a y+b x \right )^{2}-\beta \left (a y+b x \right )=-b x +\gamma } \] |
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program solution |
\[ y = \frac {\left (-2 \alpha b x +\sqrt {\frac {\left (-4 \alpha \gamma +\beta ^{2}\right ) a -4 b c \alpha }{a}}-\beta \right ) \left (x a +c \right )^{-\frac {\sqrt {\frac {\left (-4 \alpha \gamma +\beta ^{2}\right ) a -4 b c \alpha }{a}}}{2}}-2 \left (x a +c \right )^{\frac {\sqrt {\frac {\left (-4 \alpha \gamma +\beta ^{2}\right ) a -4 b c \alpha }{a}}}{2}} c_{3} \left (\alpha b x +\frac {\beta }{2}+\frac {\sqrt {\frac {\left (-4 \alpha \gamma +\beta ^{2}\right ) a -4 b c \alpha }{a}}}{2}\right )}{2 \alpha a \left (\left (x a +c \right )^{\frac {\sqrt {\frac {\left (-4 \alpha \gamma +\beta ^{2}\right ) a -4 b c \alpha }{a}}}{2}} c_{3} +\left (x a +c \right )^{-\frac {\sqrt {\frac {\left (-4 \alpha \gamma +\beta ^{2}\right ) a -4 b c \alpha }{a}}}{2}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-2 a^{2} \alpha x b -a^{2} \beta +\sqrt {-\left (\left (-4 \gamma \alpha +\beta ^{2}\right ) a -4 \alpha b c \right ) a^{3}}\, \tan \left (\frac {-2 c_{1} a^{2}+\ln \left (a x +c \right ) \sqrt {-\left (\left (-4 \gamma \alpha +\beta ^{2}\right ) a -4 \alpha b c \right ) a^{3}}}{2 a^{2}}\right )}{2 a^{3} \alpha } \] |
Problem 10376
| ODE |
\[ \boxed {2 x^{2} y^{\prime }-2 y^{2}-x y=-2 a^{2} x} \] |
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program solution |
\[ y = \frac {\left (c_{3} \cosh \left (\frac {2 a}{\sqrt {x}}\right )+\sinh \left (\frac {2 a}{\sqrt {x}}\right )\right ) a \sqrt {x}}{c_{3} \sinh \left (\frac {2 a}{\sqrt {x}}\right )+\cosh \left (\frac {2 a}{\sqrt {x}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tanh \left (\frac {i c_{1} \sqrt {x}+2 a}{\sqrt {x}}\right ) \sqrt {x}\, a \] |
Problem 10377
| ODE |
\[ \boxed {2 x^{2} y^{\prime }-2 y^{2}-3 x y=-2 a^{2} x} \] |
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program solution |
\[ y = \frac {\sinh \left (\frac {2 a}{\sqrt {x}}\right ) \left (2 a \sqrt {x}-c_{3} x \right )+2 \cosh \left (\frac {2 a}{\sqrt {x}}\right ) \left (\sqrt {x}\, c_{3} a -\frac {x}{2}\right )}{2 c_{3} \sinh \left (\frac {2 a}{\sqrt {x}}\right )+2 \cosh \left (\frac {2 a}{\sqrt {x}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (-2 x c_{1} \sqrt {-\frac {a^{2}}{x}}-x \right ) \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )-x \left (c_{1} -2 \sqrt {-\frac {a^{2}}{x}}\right ) \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right )}{2 \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right ) c_{1} +2 \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )} \] |
Problem 10378
| ODE |
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-y b x=c} \] |
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program solution |
\[ y = \frac {\left (-1-b +\sqrt {-4 c a +b^{2}+2 b +1}\right ) x^{-\frac {\sqrt {-4 c a +b^{2}+2 b +1}}{2}}-x^{\frac {\sqrt {-4 c a +b^{2}+2 b +1}}{2}} c_{3} \left (1+b +\sqrt {-4 c a +b^{2}+2 b +1}\right )}{2 x a \left (x^{\frac {\sqrt {-4 c a +b^{2}+2 b +1}}{2}} c_{3} +x^{-\frac {\sqrt {-4 c a +b^{2}+2 b +1}}{2}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-1-b +\tan \left (\frac {\sqrt {4 a c -b^{2}-2 b -1}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a c -b^{2}-2 b -1}}{2 a x} \] |
Problem 10379
| ODE |
\[ \boxed {x^{2} y^{\prime }-y^{2} c \,x^{2}-\left (x^{2} a +b x \right ) y=\alpha \,x^{2}+\beta x +\gamma } \] |
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program solution |
\[ y = -\frac {-c_{3} \left (a b -2 \beta c -\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) \sqrt {a^{2}-4 \alpha c}+\left (c_{3} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (x a +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, c x \left (c_{3} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-a b +2 \beta c +\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (a x +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) c x} \] |
Problem 10380
| ODE |
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-y b x=c \,x^{n}+s} \] |
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program solution |
\[ y = \frac {2 \sqrt {c a}\, \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}+1, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}+1, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )\right ) x^{\frac {n}{2}}-\left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )\right ) \left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right )}{2 a x \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {c a}\, x^{\frac {n}{2}}}{n}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}+1, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}+1, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \sqrt {a c}\, x^{\frac {n}{2}}-\left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right )}{2 x a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \] |
Problem 10381
| ODE |
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-y b x=c \,x^{2 n}+s \,x^{n}} \] |
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program solution |
\[ y = \frac {c_{3} \left (i \sqrt {a}\, \sqrt {c}\, s -c \left (1+b +n \right )\right ) \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, s}{2 n \sqrt {c}}+1, \frac {1+b}{2 n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right )+2 \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, s}{2 n \sqrt {c}}+1, \frac {1+b}{2 n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right ) c n -\left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, s}{2 n \sqrt {c}}, \frac {1+b}{2 n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right )+\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, s}{2 n \sqrt {c}}, \frac {1+b}{2 n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right ) c_{3} \right ) \left (2 i \sqrt {a}\, c^{\frac {3}{2}} x^{n}+i \sqrt {a}\, \sqrt {c}\, s +c \left (b -n +1\right )\right )}{2 a c x \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, s}{2 n \sqrt {c}}, \frac {1+b}{2 n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right )+\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, s}{2 n \sqrt {c}}, \frac {1+b}{2 n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\operatorname {KummerM}\left (\frac {\left (b -n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) \left (\left (-b -n -1\right ) \sqrt {c}+i \sqrt {a}\, s \right )+2 \sqrt {c}\, \operatorname {KummerU}\left (\frac {\left (b -n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) c_{1} n -2 \left (\operatorname {KummerU}\left (\frac {\left (b +n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) c_{1} +\operatorname {KummerM}\left (\frac {\left (b +n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right )\right ) \left (\frac {\left (b -n +1\right ) \sqrt {c}}{2}+i \sqrt {a}\, \left (c \,x^{n}+\frac {s}{2}\right )\right )}{2 \sqrt {c}\, x a \left (\operatorname {KummerU}\left (\frac {\left (b +n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) c_{1} +\operatorname {KummerM}\left (\frac {\left (b +n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right )\right )} \] |
Problem 10382
| ODE |
\[ \boxed {x^{2} y^{\prime }-y^{2} c \,x^{2}-\left (x^{n} a +b \right ) x y=\alpha \,x^{2 n}+\beta \,x^{n}+\gamma } \] |
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program solution |
\[ y = -\frac {\left (-x^{\frac {n}{2}} c_{3} \left (-\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-\sqrt {a^{2}-4 \alpha c}\, n +\left (b -n +1\right ) a -2 \beta c \right ) \operatorname {WhittakerM}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +\left (b -n +1\right ) a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )-2 n \,x^{\frac {n}{2}} \sqrt {a^{2}-4 \alpha c}\, \operatorname {WhittakerW}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +\left (b -n +1\right ) a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )+\left (c_{3} \operatorname {WhittakerM}\left (-\frac {\left (b -n +1\right ) a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )+\operatorname {WhittakerW}\left (-\frac {\left (b -n +1\right ) a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) \left (\left (\left (b -n +1\right ) x^{\frac {n}{2}}+a \,x^{\frac {3 n}{2}}\right ) \sqrt {a^{2}-4 \alpha c}+\left (\left (b -n +1\right ) a -2 \beta c \right ) x^{\frac {n}{2}}+x^{\frac {3 n}{2}} \left (a^{2}-4 \alpha c \right )\right )\right ) x^{-\frac {n}{2}-1}}{2 \sqrt {a^{2}-4 \alpha c}\, c \left (c_{3} \operatorname {WhittakerM}\left (-\frac {\left (b -n +1\right ) a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )+\operatorname {WhittakerW}\left (-\frac {\left (b -n +1\right ) a -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}+\sqrt {a^{2}-4 \alpha c}\, n +\left (n -b -1\right ) a +2 \beta c \right ) \operatorname {WhittakerM}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )-2 \operatorname {WhittakerW}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} n \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) \left (\left (a \,x^{n}+b -n +1\right ) \sqrt {a^{2}-4 \alpha c}+\left (a^{2}-4 \alpha c \right ) x^{n}+a \left (b -n +1\right )-2 \beta c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) c x} \] |
Problem 10383
| ODE |
\[ \boxed {x^{2} y^{\prime }-\left (\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \right ) y^{2}-\left (x^{n} a +b \right ) x y=c \,x^{2}} \] |
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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 10384
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }+\lambda \left (y^{2}-2 x y+1\right )=0} \] |
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program solution |
\[ y = \frac {\operatorname {LegendreQ}\left (\lambda , x\right )+\operatorname {LegendreP}\left (\lambda , x\right ) c_{3}}{\operatorname {LegendreP}\left (\lambda -1, x\right ) c_{3} +\operatorname {LegendreQ}\left (\lambda -1, x\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {2 \left (\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 \lambda } \left (1+x \right ) \left (-1+x \right )^{2} \operatorname {HeunCPrime}\left (0, 2 \lambda -1, 0, 0, \lambda ^{2}-\lambda +\frac {1}{2}, \frac {2}{1+x}\right )+8 \left (-1+x \right )^{2} \operatorname {HeunCPrime}\left (0, -2 \lambda +1, 0, 0, \lambda ^{2}-\lambda +\frac {1}{2}, \frac {2}{1+x}\right ) c_{1} -8 \left (1+x \right )^{2} \left (\left (\left (\lambda -\frac {1}{2}\right ) x -\frac {\lambda }{2}+\frac {1}{2}\right ) c_{1} \left (\frac {1+x}{-1+x}\right )^{-\lambda } \operatorname {hypergeom}\left (\left [1-\lambda , 1-\lambda \right ], \left [-2 \lambda +2\right ], -\frac {2}{-1+x}\right )+\frac {\lambda \left (\frac {1+x}{-1+x}\right )^{\lambda } \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 \lambda } \operatorname {hypergeom}\left (\left [\lambda , \lambda \right ], \left [2 \lambda \right ], -\frac {2}{-1+x}\right ) \left (-1+x \right )}{16}\right )\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda } \left (\frac {1+x}{-1+x}\right )^{\lambda }}{\left (8 c_{1} \operatorname {hypergeom}\left (\left [1-\lambda , 1-\lambda \right ], \left [-2 \lambda +2\right ], -\frac {2}{-1+x}\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 \lambda }+\operatorname {hypergeom}\left (\left [\lambda , \lambda \right ], \left [2 \lambda \right ], -\frac {2}{-1+x}\right ) \left (\frac {1+x}{-1+x}\right )^{2 \lambda } \left (-1+x \right )\right ) \left (1+x \right )^{2} \lambda } \] |
Problem 10385
| ODE |
\[ \boxed {\left (x^{2} a +b \right ) y^{\prime }+\alpha y^{2}+\beta x y=-\frac {b \left (a +\beta \right )}{\alpha }} \] |
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program solution |
\[ y = -\frac {\left (a +\beta \right ) x}{\alpha } \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {b \,a^{2} \left (-\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}+b \right ) \left (a \,x^{2}+2 \sqrt {-a b}\, x -b \right ) \operatorname {HeunCPrime}\left (0, -1-\frac {\beta }{a}, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta }{2 a}+\frac {\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-a b}}{-a x +\sqrt {-a b}}\right )}{2}-2 c_{1} b \left (\left (3 a \,x^{2}-b \right ) \sqrt {-a b}+x a \left (a \,x^{2}-3 b \right )\right ) a \operatorname {HeunCPrime}\left (0, \frac {\beta }{a}+1, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta }{2 a}+\frac {\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-a b}}{-a x +\sqrt {-a b}}\right )+\left (a \,x^{2}+b \right ) \left (\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}-2 \sqrt {-a b}\, x -b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )}{4}+c_{1} \left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} \left (\left (-a^{2} x^{2}+\left (-x^{2} \beta -2 b \right ) a -b \beta \right ) \sqrt {-a b}+a^{2} b x \right )\right )\right )}{\left (-\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \sqrt {-a b}\, \left (a \,x^{2}+b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )}{4}+\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} a^{2} b c_{1} \left (-\sqrt {-a b}\, x +b \right )\right ) \left (a x -\sqrt {-a b}\right )^{2} \alpha } \] |
Problem 10386
| ODE |
\[ \boxed {\left (x^{2} a +b \right ) y^{\prime }+\alpha y^{2}+\beta x y=-\gamma } \] |
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program solution |
\[ y = -\frac {\left (\operatorname {LegendreP}\left (\frac {2 a +\beta }{2 a}, \frac {\sqrt {4 \alpha \gamma a +b \,\beta ^{2}}}{2 a \sqrt {b}}, \frac {a x}{\sqrt {-a b}}\right ) c_{3} +\operatorname {LegendreQ}\left (\frac {2 a +\beta }{2 a}, \frac {\sqrt {4 \alpha \gamma a +b \,\beta ^{2}}}{2 a \sqrt {b}}, \frac {a x}{\sqrt {-a b}}\right )\right ) \left (\left (-2 a -\beta \right ) \sqrt {b}+\sqrt {4 \alpha \gamma a +b \,\beta ^{2}}\right ) \sqrt {-a b}+2 a x \sqrt {b}\, \left (\operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {\sqrt {4 \alpha \gamma a +b \,\beta ^{2}}}{2 a \sqrt {b}}, \frac {a x}{\sqrt {-a b}}\right )+\operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {\sqrt {4 \alpha \gamma a +b \,\beta ^{2}}}{2 a \sqrt {b}}, \frac {a x}{\sqrt {-a b}}\right ) c_{3} \right ) \left (a +\beta \right )}{2 \sqrt {b}\, a \alpha \left (\operatorname {LegendreQ}\left (\frac {\beta }{2 a}, \frac {\sqrt {4 \alpha \gamma a +b \,\beta ^{2}}}{2 a \sqrt {b}}, \frac {a x}{\sqrt {-a b}}\right )+\operatorname {LegendreP}\left (\frac {\beta }{2 a}, \frac {\sqrt {4 \alpha \gamma a +b \,\beta ^{2}}}{2 a \sqrt {b}}, \frac {a x}{\sqrt {-a b}}\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {b \left (a x -\sqrt {-a b}\right )^{2} \gamma \left (-2 \left (\frac {a x -\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{-\frac {\beta }{2 a}} \operatorname {hypergeom}\left (\left [\frac {2 a -\beta }{2 a}, \frac {\sqrt {4 \gamma \alpha a b +\beta ^{2} b^{2}}+\left (2 a -\beta \right ) b}{2 a b}\right ], \left [\frac {2 a -\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right ) c_{1} \sqrt {-a b}+\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} \operatorname {hypergeom}\left (\left [\frac {\beta }{2 a}, \frac {b \beta +\sqrt {4 \gamma \alpha a b +\beta ^{2} b^{2}}}{2 a b}\right ], \left [\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right ) \left (a x +\sqrt {-a b}\right )\right )}{2 \left (-a b \left (-\sqrt {-a b}\, x +b \right ) \left (a \,x^{2}+b \right ) \operatorname {HeunCPrime}\left (0, \frac {-a +\beta }{a}, -\frac {\sqrt {4 \gamma \alpha a b +\beta ^{2} b^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 \beta a +\beta ^{2}}{4 a^{2}}, -\frac {2 \sqrt {-a b}}{a x -\sqrt {-a b}}\right )+c_{1} \left (\frac {a x -\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (\left (\frac {a \,x^{2}}{2}-\sqrt {-a b}\, x -\frac {b}{2}\right ) \sqrt {4 \gamma \alpha a b +\beta ^{2} b^{2}}+a x \left (b +x^{2} \left (a -\beta \right )\right ) \sqrt {-a b}+b \left (\left (a -\frac {\beta }{2}\right ) b +a \,x^{2} \left (a -\frac {3 \beta }{2}\right )\right )\right ) b \left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{-\frac {\beta }{2 a}} \operatorname {hypergeom}\left (\left [\frac {2 a -\beta }{2 a}, \frac {\sqrt {4 \gamma \alpha a b +\beta ^{2} b^{2}}+\left (2 a -\beta \right ) b}{2 a b}\right ], \left [\frac {2 a -\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )+2 a \left (\frac {a x -\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} b^{2} c_{1} \left (a \,x^{2}+2 \sqrt {-a b}\, x -b \right ) \operatorname {HeunCPrime}\left (0, \frac {a -\beta }{a}, -\frac {\sqrt {4 \gamma \alpha a b +\beta ^{2} b^{2}}}{2 a b}, 0, \frac {2 a^{2}-2 \beta a +\beta ^{2}}{4 a^{2}}, -\frac {2 \sqrt {-a b}}{a x -\sqrt {-a b}}\right )+\frac {\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} \operatorname {hypergeom}\left (\left [\frac {\beta }{2 a}, \frac {b \beta +\sqrt {4 \gamma \alpha a b +\beta ^{2} b^{2}}}{2 a b}\right ], \left [\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right ) \left (\sqrt {-a b}\, x +b \right ) \left (-b \beta +\sqrt {4 \gamma \alpha a b +\beta ^{2} b^{2}}\right ) \left (a \,x^{2}+b \right )}{4}\right ) a} \] |
Problem 10387
| ODE |
\[ \boxed {\left (x^{2} a +b \right ) y^{\prime }+y^{2}-2 x y=-\left (1-a \right ) x^{2}+b} \] |
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program solution |
\[ y = \frac {\arctan \left (\frac {\sqrt {a b}\, x}{b}\right ) x +\sqrt {a b}+c_{3} x}{\arctan \left (\frac {\sqrt {a b}\, x}{b}\right )+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x +\frac {\sqrt {a b}}{c_{1} \sqrt {a b}+\arctan \left (\frac {a x}{\sqrt {a b}}\right )} \] |
Problem 10388
| ODE |
\[ \boxed {\left (x^{2} a +b x +c \right ) y^{\prime }-y^{2}-\left (2 \lambda x +b \right ) y=\lambda \left (\lambda -a \right ) x^{2}+\mu } \] |
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program solution |
\[ y = -\frac {8 \left (x^{2} a +b x +c \right )^{2} a^{2} \left (\left (i a \sqrt {4 c a -b^{2}}\, \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}-\sqrt {-4 c a +b^{2}}\, \left (2 \lambda x +b \right )\right ) {\left (\frac {-b +i \sqrt {4 c a -b^{2}}-2 x a}{i \sqrt {4 c a -b^{2}}+2 x a +b}\right )}^{-\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}}-{\left (\frac {-b +i \sqrt {4 c a -b^{2}}-2 x a}{i \sqrt {4 c a -b^{2}}+2 x a +b}\right )}^{\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} \left (i a \sqrt {4 c a -b^{2}}\, \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}+\sqrt {-4 c a +b^{2}}\, \left (2 \lambda x +b \right )\right ) c_{3} \right )}{\sqrt {-4 c a +b^{2}}\, \left (2 x a +b -\sqrt {-4 c a +b^{2}}\right ) \left (2 x a +b +\sqrt {-4 c a +b^{2}}\right ) \left (i \sqrt {4 c a -b^{2}}+2 x a +b \right ) \left (-b +i \sqrt {4 c a -b^{2}}-2 x a \right ) \left (c_{3} {\left (\frac {-b +i \sqrt {4 c a -b^{2}}-2 x a}{i \sqrt {4 c a -b^{2}}+2 x a +b}\right )}^{\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}}+{\left (\frac {-b +i \sqrt {4 c a -b^{2}}-2 x a}{i \sqrt {4 c a -b^{2}}+2 x a +b}\right )}^{-\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {8 \left (\left (i a \sqrt {4 a c -b^{2}}\, \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}-\sqrt {-4 a c +b^{2}}\, \left (2 x \lambda +b \right )\right ) c_{1} {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-\left (i a \sqrt {4 a c -b^{2}}\, \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}+\sqrt {-4 a c +b^{2}}\, \left (2 x \lambda +b \right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right ) \left (a \,x^{2}+b x +c \right )^{2} a^{2}}{\sqrt {-4 a c +b^{2}}\, \left (2 a x -\sqrt {-4 a c +b^{2}}+b \right ) \left (2 a x +\sqrt {-4 a c +b^{2}}+b \right ) \left (i \sqrt {4 a c -b^{2}}+2 a x +b \right ) \left (-b +i \sqrt {4 a c -b^{2}}-2 a x \right ) \left (c_{1} {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )} \] |
Problem 10389
| ODE |
\[ \boxed {\left (x^{2} a +b x +c \right ) y^{\prime }-y^{2}-\left (x a +\mu \right ) y=-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +c \lambda } \] |
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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 10390
| ODE |
\[ \boxed {\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime }-y^{2}-\left (a_{1} x +b_{1} \right ) y=-\lambda \left (\lambda +a_{1} -a_{2} \right ) x^{2}+\lambda \left (b_{2} -b_{1} \right ) x +\lambda c_{2}} \] |
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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 10391
| ODE |
\[ \boxed {\left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime }-y^{2}-\left (a_{1} x +b_{1} \right ) y=a_{0} x^{2}+b_{0} x +c_{0}} \] |
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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 10392
| ODE |
\[ \boxed {\left (x -a \right ) \left (x -b \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right )=0} \] |
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program solution |
\[ y = \frac {k \left (a -x \right )^{k +1} \left (b -x \right )^{k +1} \left (c_{3} \left (a -x \right )^{-k -1}+\left (b -x \right )^{-k -1}\right )}{\left (k +1\right ) \left (c_{3} \left (b -x \right )^{k}+\left (a -x \right )^{k}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {k \left (\left (b -x \right )^{1+k}+c_{1} \left (a -x \right )^{k} \left (a -x \right )\right )}{\left (1+k \right ) \left (c_{1} \left (a -x \right )^{k}+\left (b -x \right )^{k}\right )} \] |
Problem 10393
| ODE |
\[ \boxed {\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b_{1} x +a_{1} \right ) y=-a_{0}} \] |
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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 10394
| ODE |
\[ \boxed {x^{3} y^{\prime }-a \,x^{3} y^{2}-\left (b \,x^{2}+c \right ) y=s x} \] |
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program solution |
\[ y = -\frac {8 \left (\frac {c_{3} \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{4}+\operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right ) x \,s^{2} a}{\left (\frac {\left (\left (-1+b \right ) x^{2}+c \right ) \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) c_{3} \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+x^{2} c_{3} \left (\frac {1}{2}+\frac {\left (-1+b \right ) \sqrt {-4 a s +b^{2}+2 b +1}}{2}+a s -\frac {b^{2}}{2}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+2 \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) \left (\left (-1+b \right ) x^{2}+c \right )-4 \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) x^{2}\right ) \left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right ) \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (\frac {\left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{4}+c_{1} \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right ) \left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right ) x \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right )}{2 a \left (\frac {\left (\left (b -1\right ) x^{2}+c \right ) \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+\left (\frac {1}{2}+\frac {\left (b -1\right ) \sqrt {-4 a s +b^{2}+2 b +1}}{2}+a s -\frac {b^{2}}{2}\right ) x^{2} \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )-4 c_{1} \left (\frac {\left (\left (-b +1\right ) x^{2}-c \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+\operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) x^{2}\right )\right )} \] |
Problem 10395
| ODE |
\[ \boxed {x^{3} y^{\prime }-a \,x^{3} y^{2}-x \left (b x +c \right ) y=\alpha x +\beta } \] |
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program solution |
\[ y = \frac {\left (\left (\alpha a +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (-c_{3} \left (-c^{2}-x \left (2+b \right ) c +\beta a x \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (c^{2}+x \left (2+b \right ) c -\beta a x \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )+\left (-\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right ) c_{3} x \operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )\right )}{a \,c^{2} x^{2} \left (\operatorname {KummerM}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{3} +\operatorname {KummerU}\left (\frac {\sqrt {-4 \alpha a +b^{2}+2 b +1}\, c +\left (b +3\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 \alpha a +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (\left (a \alpha +b +2\right ) c^{2}-a \beta \left (b +3\right ) c +a^{2} \beta ^{2}\right ) x c_{1} \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c \left (\left (c^{2}+x \left (b +2\right ) c -a x \beta \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )-c_{1} \left (-c^{2}-x \left (b +2\right ) c +a x \beta \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )+x \operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +\left (b +5\right ) c -2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) \left (-\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c}{2}+\frac {\left (-b -3\right ) c}{2}+\beta a \right )\right )}{c^{2} x^{2} a \left (\operatorname {KummerU}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right ) c_{1} +\operatorname {KummerM}\left (\frac {\sqrt {-4 a \alpha +b^{2}+2 b +1}\, c +c \left (b +3\right )-2 \beta a}{2 c}, 1+\sqrt {-4 a \alpha +b^{2}+2 b +1}, \frac {c}{x}\right )\right )} \] |
Problem 10396
| ODE |
\[ \boxed {x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y=-s x} \] |
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program solution |
\[ y = \frac {\left (a -\frac {c}{3}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (-3+b \right ) a -c^{2}\right ) \left (a \,x^{\frac {a +c}{a}}+x^{\frac {3 a +c}{a}}\right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -7 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +7\right ) a +2 c}{4 a}\right ], \left [\frac {3 a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (-\left (\left (-2+b \right ) a -c \right ) \left (a -\frac {c}{3}\right ) x^{\frac {a +c}{a}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (\frac {2 \left (x^{2}+a \right ) x^{2} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {5 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right ) \left (-\frac {\lambda s}{2}+b -3\right )}{3}+\operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right ) \left (a -\frac {c}{3}\right ) \left (a -c +\left (-b +3\right ) x^{2}\right )\right ) c_{3} a \right ) \left (a +c \right ) a}{\left (a +c \right ) \left (a -\frac {c}{3}\right ) \left (x^{2}+a \right ) \left (x \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_{3} +\operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right ) x^{\frac {c}{a}}\right ) \lambda \,a^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (a -\frac {c}{3}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (b -3\right ) a -c^{2}\right ) \left (a \,x^{\frac {a +c}{a}}+x^{\frac {3 a +c}{a}}\right ) c_{1} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -7 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +7\right ) a +2 c}{4 a}\right ], \left [\frac {3 a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (-\left (a -\frac {c}{3}\right ) \left (\left (b -2\right ) a -c \right ) c_{1} x^{\frac {a +c}{a}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (\frac {2 \left (x^{2}+a \right ) x^{2} \left (-\frac {\lambda s}{2}+b -3\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {5 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{3}+\left (a -\frac {c}{3}\right ) \left (a -c +\left (-b +3\right ) x^{2}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )\right ) a \right ) \left (a +c \right ) a}{\left (a -\frac {c}{3}\right ) \left (x \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_{1} x^{\frac {c}{a}}\right ) \left (x^{2}+a \right ) \left (a +c \right ) a^{2} \lambda } \] |
Problem 10397
| ODE |
\[ \boxed {x^{2} \left (x +a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b x +c \right ) y=-\alpha x -\beta } \] |
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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 10398
| ODE |
\[ \boxed {\left (x^{2} a +b x +e \right ) \left (y^{\prime } x -y\right )-y^{2}=-x^{2}} \] |
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program solution |
\[ y = -\frac {\left (c_{3} \cosh \left (\frac {2 \arctan \left (\frac {2 x a +b}{\sqrt {4 a e -b^{2}}}\right )}{\sqrt {4 a e -b^{2}}}\right )+\sinh \left (\frac {2 \arctan \left (\frac {2 x a +b}{\sqrt {4 a e -b^{2}}}\right )}{\sqrt {4 a e -b^{2}}}\right )\right ) x}{c_{3} \sinh \left (\frac {2 \arctan \left (\frac {2 x a +b}{\sqrt {4 a e -b^{2}}}\right )}{\sqrt {4 a e -b^{2}}}\right )+\cosh \left (\frac {2 \arctan \left (\frac {2 x a +b}{\sqrt {4 a e -b^{2}}}\right )}{\sqrt {4 a e -b^{2}}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\tanh \left (\frac {c_{1} \sqrt {4 e a -b^{2}}+2 \arctan \left (\frac {2 a x +b}{\sqrt {4 e a -b^{2}}}\right )}{\sqrt {4 e a -b^{2}}}\right ) x \] |
Problem 10399
| ODE |
\[ \boxed {x^{2} \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+x \left (b \,x^{2}+c \right ) y=-s} \] |
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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 10400
| ODE |
\[ \boxed {a \left (x^{2}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+b x \left (x^{2}-1\right ) y=-c \,x^{2}-d x -s} \] |
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program solution |
\[ y = \frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) a \left (x^{2}-1\right )+b \left (x^{3}-x \right ) \textit {\_Y}^{\prime }\left (x \right )+\left (c \,x^{2}+d x +s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\lambda \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) a \left (x^{2}-1\right )+b \left (x^{3}-x \right ) \textit {\_Y}^{\prime }\left (x \right )+\left (c \,x^{2}+d x +s \right ) \lambda \textit {\_Y} \left (x \right )}{a \left (x^{2}-1\right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |