Problem 9901
| ODE |
\[ \boxed {y^{\left (5\right )}+2 y^{\prime \prime \prime }+y^{\prime }=x a +b \sin \left (x \right )+c \cos \left (x \right )} \] |
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program solution |
\[ y = \left (c_{5} x +c_{4} \right ) {\mathrm e}^{-i x}+\left (c_{3} x +c_{2} \right ) {\mathrm e}^{i x}+c_{1} -\frac {\left (\int \left (x a +b \sin \left (x \right )+c \cos \left (x \right )\right ) \left (2 \cos \left (x \right )+x \sin \left (x \right )\right )d x \right ) \cos \left (x \right )}{2}+\frac {\left (\int \left (x a +b \sin \left (x \right )+c \cos \left (x \right )\right ) \left (-2 \sin \left (x \right )+\cos \left (x \right ) x \right )d x \right ) \sin \left (x \right )}{2}+\frac {\left (4 b \,x^{2}+x \left (i b -c \right )-16 b \right ) \cos \left (x \right )}{16}+\frac {\left (-4 c \,x^{2}+\left (i c -3 b \right ) x +16 c \right ) \sin \left (x \right )}{16} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (b \,x^{2}+\left (-4 c -8 c_{4} \right ) x -6 b -8 c_{2} +8 c_{3} \right ) \cos \left (x \right )}{8}+\frac {\left (-x^{2} c +\left (-4 b +8 c_{3} \right ) x +6 c +8 c_{1} +8 c_{4} \right ) \sin \left (x \right )}{8}+\frac {a \,x^{2}}{2}+c_{5} \] |
Problem 9902
| ODE |
\[ \boxed {y^{\left (6\right )}+y=\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} c_{3} +{\mathrm e}^{-i x} c_{4} +{\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} c_{5} +{\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} c_{6} +\frac {\cos \left (x \right )}{6}-\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sin \left (x \right )d x \right ) \cos \left (x \right )}{3}+\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \cos \left (x \right )d x \right ) \sin \left (x \right )}{3}-\frac {\cos \left (2 x \right )}{21} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\sin \left (\frac {x}{2}\right ) c_{4} +\cos \left (\frac {x}{2}\right ) c_{3} \right ) {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}+\left (\sin \left (\frac {x}{2}\right ) c_{6} +c_{5} \cos \left (\frac {x}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}+\frac {\cos \left (2 x \right )}{126}+\frac {\left (5+24 c_{1} \right ) \cos \left (x \right )}{24}+\frac {\sin \left (x \right ) \left (x +12 c_{2} \right )}{12} \] |
Problem 9903
| ODE |
\[ \boxed {y^{\left (5\right )}-y a x=b} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9904
| ODE |
\[ \boxed {y^{\left (5\right )}+a \,x^{\nu } y^{\prime }+a \nu \,x^{\nu -1} y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9905
| ODE |
\[ \boxed {y^{\left (5\right )}+a y^{\prime \prime \prime \prime }=f} \] |
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program solution |
\[ y = c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{-x a} c_{5} +\frac {f \,x^{4}}{24 a} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {6 \,{\mathrm e}^{-a x} c_{1} +a^{3} \left (\left (c_{2} x^{3}+3 x^{2} c_{3} +6 c_{4} x +6 c_{5} \right ) a +\frac {f \,x^{4}}{4}\right )}{6 a^{4}} \] |
Problem 9906
| ODE |
\[ \boxed {x y^{\left (5\right )}-m n y^{\prime \prime \prime \prime }+y a x=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \frac {1}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{5}, \frac {4}{5}, \frac {6}{5}, \frac {2}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{5}, \frac {6}{5}, \frac {7}{5}, \frac {3}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{4} x^{3} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}, \frac {7}{5}, \frac {8}{5}, \frac {4}{5}-\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right )+c_{5} x^{m n +4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}+\frac {m n}{5}, \frac {9}{5}+\frac {m n}{5}, \frac {8}{5}+\frac {m n}{5}, \frac {7}{5}+\frac {m n}{5}\right ], -\frac {a \,x^{5}}{3125}\right ) \] |
Problem 9907
| ODE |
Solve \begin {gather*} \boxed {x \left (y^{\prime } a +b y^{\prime \prime }+c y^{\prime \prime \prime }+e y^{\prime \prime \prime \prime }\right ) y=0} \end {gather*} |
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program solution |
N/A |
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= {\mathrm e}^{\frac {\left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {2}{3}}-2 c \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}-12 b e +4 c^{2}\right ) x}{6 e \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}}} c_{4} +{\mathrm e}^{\frac {x \left (i \left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {2}{3}}+12 b e -4 c^{2}\right ) \sqrt {3}+12 b e -{\left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}+2 c \right )}^{2}\right )}{12 e \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}}} c_{3} +c_{2} {\mathrm e}^{\frac {\left (-i \left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {2}{3}}+12 b e -4 c^{2}\right ) \sqrt {3}+12 b e -{\left (\left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}+2 c \right )}^{2}\right ) x}{12 e \left (12 \sqrt {3}\, \sqrt {27 a^{2} e^{2}+\left (-18 a c b +4 b^{3}\right ) e +4 a \,c^{3}-b^{2} c^{2}}\, e -108 a \,e^{2}+36 b c e -8 c^{3}\right )^{\frac {1}{3}}}}+c_{1} \\ \end{align*} |
Problem 9908
| ODE |
\[ \boxed {x y^{\left (5\right )}-\left (\left (a A_{2} -A_{1} \right ) x +A_{2} \right ) y^{\prime }=\left (a A_{1} -A_{0} \right ) x +A_{1}} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9909
| ODE |
\[ \boxed {x^{2} y^{\prime \prime \prime \prime }-a y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\left (\left (\operatorname {BesselY}\left (1, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_{4} +\operatorname {BesselJ}\left (1, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_{3} \right ) a^{\frac {1}{4}}+\left (\operatorname {BesselJ}\left (1, 2 a^{\frac {1}{4}} \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (1, 2 a^{\frac {1}{4}} \sqrt {x}\right ) c_{2} \right ) \sqrt {-\sqrt {a}}\right ) \sqrt {x}-\left (\operatorname {BesselY}\left (0, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_{4} +\operatorname {BesselJ}\left (0, 2 \sqrt {-\sqrt {a}}\, \sqrt {x}\right ) c_{3} +\operatorname {BesselY}\left (0, 2 a^{\frac {1}{4}} \sqrt {x}\right ) c_{2} +\operatorname {BesselJ}\left (0, 2 a^{\frac {1}{4}} \sqrt {x}\right ) c_{1} \right ) \sqrt {-\sqrt {a}}\, a^{\frac {1}{4}} x}{\sqrt {-\sqrt {a}}\, a^{\frac {1}{4}}} \] |
Problem 9910
| ODE |
\[ \boxed {x^{10} y^{\left (5\right )}-a y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {6}{5}, \frac {7}{5}, \frac {8}{5}, \frac {9}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{2} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {4}{5}, \frac {6}{5}, \frac {7}{5}, \frac {8}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{3} x^{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{5}, \frac {4}{5}, \frac {6}{5}, \frac {7}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{4} x^{3} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2}{5}, \frac {3}{5}, \frac {4}{5}, \frac {6}{5}\right ], -\frac {a}{3125 x^{5}}\right )+c_{5} x^{4} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{5}, \frac {2}{5}, \frac {3}{5}, \frac {4}{5}\right ], -\frac {a}{3125 x^{5}}\right ) \] |
Problem 9911
| ODE |
\[ \boxed {x^{\frac {5}{2}} y^{\left (5\right )}-a y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {8 \left (\left (x^{\frac {3}{2}} \left (\left (c_{2} -c_{5} \right ) \cos \left (\frac {\pi }{5}\right )+\left (-c_{3} +\frac {c_{4}}{15}\right ) \cos \left (\frac {2 \pi }{5}\right )+c_{1} \right ) a^{\frac {2}{5}}+\left (\frac {3 \left (c_{1} +\frac {c_{4}}{15}\right ) x \,a^{\frac {1}{5}}}{2}+\frac {3 \sqrt {x}\, \left (c_{2} +c_{3} \right )}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\left (\frac {3 x \left (c_{3} -c_{5} \right ) a^{\frac {1}{5}}}{2}+\frac {3 \sqrt {x}\, \left (c_{1} +c_{5} \right )}{4}\right ) \cos \left (\frac {2 \pi }{5}\right )+\frac {\sqrt {x}\, c_{4}}{20}+\frac {3 x c_{2} a^{\frac {1}{5}}}{2}\right ) \cos \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}\right )+\sin \left (2 \sin \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}\right ) \left (\left (\left (c_{2} +c_{5} \right ) \sin \left (\frac {\pi }{5}\right )+\sin \left (\frac {2 \pi }{5}\right ) \left (c_{3} +\frac {c_{4}}{15}\right )\right ) x^{\frac {3}{2}} a^{\frac {2}{5}}+\left (-\frac {3 \left (c_{1} -\frac {c_{4}}{15}\right ) x \,a^{\frac {1}{5}}}{2}-\frac {3 \sqrt {x}\, \left (c_{2} -c_{3} \right )}{4}\right ) \sin \left (\frac {\pi }{5}\right )-\frac {3 \sin \left (\frac {2 \pi }{5}\right ) \left (-2 x \left (c_{3} +c_{5} \right ) a^{\frac {1}{5}}+\sqrt {x}\, \left (c_{1} -c_{5} \right )\right )}{4}\right )\right ) {\mathrm e}^{-2 \cos \left (\frac {\pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}}+8 \left (\left (x^{\frac {3}{2}} \left (\left (c_{3} -\frac {c_{4}}{15}\right ) \cos \left (\frac {\pi }{5}\right )+\left (-c_{2} +c_{5} \right ) \cos \left (\frac {2 \pi }{5}\right )+c_{1} \right ) a^{\frac {2}{5}}+\left (-\frac {3 x \left (c_{3} -c_{5} \right ) a^{\frac {1}{5}}}{2}-\frac {3 \sqrt {x}\, \left (c_{1} +c_{5} \right )}{4}\right ) \cos \left (\frac {\pi }{5}\right )+\left (-\frac {3 \left (c_{1} +\frac {c_{4}}{15}\right ) x \,a^{\frac {1}{5}}}{2}-\frac {3 \sqrt {x}\, \left (c_{2} +c_{3} \right )}{4}\right ) \cos \left (\frac {2 \pi }{5}\right )+\frac {\sqrt {x}\, c_{4}}{20}+\frac {3 x c_{2} a^{\frac {1}{5}}}{2}\right ) \cos \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}\right )-\left (x^{\frac {3}{2}} \left (\left (c_{3} +\frac {c_{4}}{15}\right ) \sin \left (\frac {\pi }{5}\right )-\sin \left (\frac {2 \pi }{5}\right ) \left (c_{2} +c_{5} \right )\right ) a^{\frac {2}{5}}+\left (\frac {3 x \left (c_{3} +c_{5} \right ) a^{\frac {1}{5}}}{2}-\frac {3 \sqrt {x}\, \left (c_{1} -c_{5} \right )}{4}\right ) \sin \left (\frac {\pi }{5}\right )+\frac {3 \sin \left (\frac {2 \pi }{5}\right ) \left (2 \left (c_{1} -\frac {c_{4}}{15}\right ) x \,a^{\frac {1}{5}}+\sqrt {x}\, \left (c_{2} -c_{3} \right )\right )}{4}\right ) \sin \left (2 \sin \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}\right )\right ) {\mathrm e}^{2 \cos \left (\frac {2 \pi }{5}\right ) \sqrt {x}\, a^{\frac {1}{5}}}+4 \left (x^{\frac {3}{2}} a^{\frac {2}{5}}-\frac {3 a^{\frac {1}{5}} x}{2}+\frac {3 \sqrt {x}}{4}\right ) \left (c_{1} -c_{2} -c_{3} +\frac {c_{4}}{15}+c_{5} \right ) {\mathrm e}^{2 \sqrt {x}\, a^{\frac {1}{5}}}}{\sqrt {x}} \] |
Problem 9912
| ODE |
\[ \boxed {\left (x -a \right )^{5} \left (x -b \right )^{5} y^{\left (5\right )}-y c=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9913
| ODE |
\[ \boxed {y^{\prime \prime }-y^{2}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {3}{\sqrt {6 \textit {\_a}^{3}+18 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {3}{\sqrt {6 \textit {\_a}^{3}+18 c_{1}}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = 6 \operatorname {WeierstrassP}\left (x +c_{1} , 0, c_{2}\right ) \] |
Problem 9914
| ODE |
\[ \boxed {y^{\prime \prime }-6 y^{2}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\sqrt {4 \textit {\_a}^{3}+2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {1}{\sqrt {4 \textit {\_a}^{3}+2 c_{1}}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \operatorname {WeierstrassP}\left (x +c_{1} , 0, c_{2}\right ) \] |
Problem 9915
| ODE |
\[ \boxed {y^{\prime \prime }-6 y^{2}=x} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9916
| ODE |
\[ \boxed {y^{\prime \prime }-6 y^{2}+4 y=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\sqrt {4 \textit {\_a}^{3}-4 \textit {\_a}^{2}+2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {1}{\sqrt {4 \textit {\_a}^{3}-4 \textit {\_a}^{2}+2 c_{1}}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {4 \textit {\_a}^{3}-4 \textit {\_a}^{2}+c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {4 \textit {\_a}^{3}-4 \textit {\_a}^{2}+c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9917
| ODE |
\[ \boxed {y^{\prime \prime }+a y^{2}=-b x -c} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9918
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{3}-x y=-a} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9919
| ODE |
\[ \boxed {y^{\prime \prime }-a y^{3}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {2}{\sqrt {2 \textit {\_a}^{4} a +8 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {2}{\sqrt {2 \textit {\_a}^{4} a +8 c_{1}}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = c_{2} \operatorname {JacobiSN}\left (\frac {\left (\sqrt {2}\, \sqrt {-a}\, x +2 c_{1} \right ) c_{2}}{2}, i\right ) \] |
Problem 9920
| ODE |
\[ \boxed {y^{\prime \prime }-2 a^{2} y^{3}+2 a b x y=b} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9921
| ODE |
\[ \boxed {y^{\prime \prime }+y b x +y c +a y^{3}=-d} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9922
| ODE |
\[ \boxed {y^{\prime \prime }+y^{2} b +y c +a y^{3}=-d} \] |
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program solution |
\[ \int _{}^{y}\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {6}{\sqrt {-18 \textit {\_a}^{4} a -24 \textit {\_a}^{3} b -36 \textit {\_a}^{2} c -72 \textit {\_a} d +72 c_{1}}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} -6 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-18 a \,\textit {\_a}^{4}-24 b \,\textit {\_a}^{3}-36 \textit {\_a}^{2} c -72 \textit {\_a} d +36 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 6 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-18 a \,\textit {\_a}^{4}-24 b \,\textit {\_a}^{3}-36 \textit {\_a}^{2} c -72 \textit {\_a} d +36 c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9923
| ODE |
\[ \boxed {y^{\prime \prime }+a \,x^{r} y^{2}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9924
| ODE |
\[ \boxed {y^{\prime \prime }+6 a^{10} y^{11}-y=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\sqrt {-\textit {\_a}^{12} a^{10}+\textit {\_a}^{2}+2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {1}{\sqrt {-\textit {\_a}^{12} a^{10}+\textit {\_a}^{2}+2 c_{1}}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\textit {\_a}^{12} a^{10}+\textit {\_a}^{2}+c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\textit {\_a}^{12} a^{10}+\textit {\_a}^{2}+c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9925
| ODE |
\[ \boxed {y^{\prime \prime }-\frac {1}{\left (a y^{2}+y b x +c \,x^{2}+\alpha y+\beta x +\gamma \right )^{\frac {3}{2}}}=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {2 \operatorname {RootOf}\left (2 \left (2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {16 \sqrt {4 a \beta +4 a c +4 a \gamma -\alpha ^{2}-2 b \alpha -b^{2}}\, a \left (\int \frac {1}{\left (4 \textit {\_g}^{2} a^{2}+1\right ) \sqrt {\frac {\left (4 a \beta +4 a c +4 a \gamma -\alpha ^{2}-2 b \alpha -b^{2}\right ) \left (4 \textit {\_g}^{2} a^{2}+1\right )}{a}}}d \textit {\_g} \right )+4 \beta ^{2} a^{2} \textit {\_g}^{2}-16 a^{2} \gamma c \,\textit {\_g}^{2}+4 c_{1} a^{2} \beta ^{2}+4 a c \,\alpha ^{2} \textit {\_g}^{2}-4 a b \beta \alpha \,\textit {\_g}^{2}+4 a \,b^{2} \gamma \,\textit {\_g}^{2}-4 c_{1} a \alpha b \beta +c_{1} \alpha ^{2} b^{2}}}d \textit {\_g} \right ) a \beta -\left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {16 \sqrt {4 a \beta +4 a c +4 a \gamma -\alpha ^{2}-2 b \alpha -b^{2}}\, a \left (\int \frac {1}{\left (4 \textit {\_g}^{2} a^{2}+1\right ) \sqrt {\frac {\left (4 a \beta +4 a c +4 a \gamma -\alpha ^{2}-2 b \alpha -b^{2}\right ) \left (4 \textit {\_g}^{2} a^{2}+1\right )}{a}}}d \textit {\_g} \right )+4 \beta ^{2} a^{2} \textit {\_g}^{2}-16 a^{2} \gamma c \,\textit {\_g}^{2}+4 c_{1} a^{2} \beta ^{2}+4 a c \,\alpha ^{2} \textit {\_g}^{2}-4 a b \beta \alpha \,\textit {\_g}^{2}+4 a \,b^{2} \gamma \,\textit {\_g}^{2}-4 c_{1} a \alpha b \beta +c_{1} \alpha ^{2} b^{2}}}d \textit {\_g} \right ) \alpha b +c_{2} \right ) \sqrt {-a \left (\beta ^{2} a -4 a c \gamma +\alpha ^{2} c -\alpha b \beta +b^{2} \gamma \right )}-\arctan \left (\frac {4 a c x -b^{2} x +2 a \beta -b \alpha }{2 \sqrt {-a \left (\beta ^{2} a -4 a c \gamma +\alpha ^{2} c -\alpha b \beta +b^{2} \gamma \right )}}\right ) \left (2 a \beta -b \alpha \right )\right ) \sqrt {4 a c \,x^{2}-b^{2} x^{2}+4 x \beta a -2 \alpha b x +4 a \gamma -\alpha ^{2}}\, a -b x -\alpha }{2 a} \\ y \left (x \right ) &= \frac {2 \operatorname {RootOf}\left (2 \left (-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {16 \sqrt {4 a \beta +4 a c +4 a \gamma -\alpha ^{2}-2 b \alpha -b^{2}}\, a \left (\int \frac {1}{\left (4 \textit {\_g}^{2} a^{2}+1\right ) \sqrt {\frac {\left (4 a \beta +4 a c +4 a \gamma -\alpha ^{2}-2 b \alpha -b^{2}\right ) \left (4 \textit {\_g}^{2} a^{2}+1\right )}{a}}}d \textit {\_g} \right )+4 \beta ^{2} a^{2} \textit {\_g}^{2}-16 a^{2} \gamma c \,\textit {\_g}^{2}+4 c_{1} a^{2} \beta ^{2}+4 a c \,\alpha ^{2} \textit {\_g}^{2}-4 a b \beta \alpha \,\textit {\_g}^{2}+4 a \,b^{2} \gamma \,\textit {\_g}^{2}-4 c_{1} a \alpha b \beta +c_{1} \alpha ^{2} b^{2}}}d \textit {\_g} \right ) a \beta +\left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {16 \sqrt {4 a \beta +4 a c +4 a \gamma -\alpha ^{2}-2 b \alpha -b^{2}}\, a \left (\int \frac {1}{\left (4 \textit {\_g}^{2} a^{2}+1\right ) \sqrt {\frac {\left (4 a \beta +4 a c +4 a \gamma -\alpha ^{2}-2 b \alpha -b^{2}\right ) \left (4 \textit {\_g}^{2} a^{2}+1\right )}{a}}}d \textit {\_g} \right )+4 \beta ^{2} a^{2} \textit {\_g}^{2}-16 a^{2} \gamma c \,\textit {\_g}^{2}+4 c_{1} a^{2} \beta ^{2}+4 a c \,\alpha ^{2} \textit {\_g}^{2}-4 a b \beta \alpha \,\textit {\_g}^{2}+4 a \,b^{2} \gamma \,\textit {\_g}^{2}-4 c_{1} a \alpha b \beta +c_{1} \alpha ^{2} b^{2}}}d \textit {\_g} \right ) \alpha b +c_{2} \right ) \sqrt {-a \left (\beta ^{2} a -4 a c \gamma +\alpha ^{2} c -\alpha b \beta +b^{2} \gamma \right )}-\arctan \left (\frac {4 a c x -b^{2} x +2 a \beta -b \alpha }{2 \sqrt {-a \left (\beta ^{2} a -4 a c \gamma +\alpha ^{2} c -\alpha b \beta +b^{2} \gamma \right )}}\right ) \left (2 a \beta -b \alpha \right )\right ) \sqrt {4 a c \,x^{2}-b^{2} x^{2}+4 x \beta a -2 \alpha b x +4 a \gamma -\alpha ^{2}}\, a -b x -\alpha }{2 a} \\ \end{align*} |
Problem 9926
| ODE |
\[ \boxed {y^{\prime \prime }-{\mathrm e}^{y}=0} \] |
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program solution |
\[ y = \ln \left (\tanh \left (\frac {\sqrt {c_{1}}\, \left (x +c_{2} \right ) \sqrt {2}}{2}\right )^{2} c_{1} -c_{1} \right ) \] Verified OK. \[ y = \ln \left (\tanh \left (\frac {\sqrt {c_{1}}\, \left (c_{3} +x \right ) \sqrt {2}}{2}\right )^{2} c_{1} -c_{1} \right ) \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = -\ln \left (2\right )+\ln \left (\frac {\sec \left (\frac {x +c_{2}}{2 c_{1}}\right )^{2}}{c_{1}^{2}}\right ) \] |
Problem 9927
| ODE |
\[ \boxed {y^{\prime \prime }+a \,{\mathrm e}^{x} \sqrt {y}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9928
| ODE |
\[ \boxed {y^{\prime \prime }+{\mathrm e}^{x} \sin \left (y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9929
| ODE |
\[ \boxed {y^{\prime \prime }+a \sin \left (y\right )=0} \] |
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program solution |
\[ \frac {2 \sqrt {\frac {a \cos \left (y\right )+c_{1}}{c_{1} +a}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {\sqrt {2}\, \sqrt {a}}{\sqrt {c_{1} +a}}\right )}{\sqrt {2 a \cos \left (y\right )+2 c_{1}}} = x +c_{2} \] Verified OK. \[ -\frac {2 \sqrt {\frac {a \cos \left (y\right )+c_{1}}{c_{1} +a}}\, \operatorname {InverseJacobiAM}\left (\frac {y}{2}, \frac {\sqrt {2}\, \sqrt {a}}{\sqrt {c_{1} +a}}\right )}{\sqrt {2 a \cos \left (y\right )+2 c_{1}}} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 a \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {2 a \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9930
| ODE |
\[ \boxed {y^{\prime \prime }+a^{2} \sin \left (y\right )=\beta \sin \left (x \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9931
| ODE |
\[ \boxed {y^{\prime \prime }+a^{2} \sin \left (y\right )=\beta f \left (x \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9932
| ODE |
\[ \boxed {y^{\prime \prime }-\frac {f \left (\frac {y}{\sqrt {x}}\right )}{x^{\frac {3}{2}}}=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (\textit {\_Z} \,x^{\frac {3}{2}}+4 f \left (\frac {\textit {\_Z}}{\sqrt {x}}\right ) x^{2}\right ) \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {c_{1} +8 \left (\int f \left (\textit {\_g} \right )d \textit {\_g} \right )+\textit {\_g}^{2}}}d \textit {\_g} \right )+2 c_{2} \right ) \sqrt {x} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {c_{1} +8 \left (\int f \left (\textit {\_g} \right )d \textit {\_g} \right )+\textit {\_g}^{2}}}d \textit {\_g} \right )+2 c_{2} \right ) \sqrt {x} \\ \end{align*} |
Problem 9933
| ODE |
\[ \boxed {y^{\prime \prime }-3 y^{\prime }-y^{2}-2 y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9934
| ODE |
\[ \boxed {y^{\prime \prime }-7 y^{\prime }-y^{\frac {3}{2}}+12 y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9935
| ODE |
\[ \boxed {y^{\prime \prime }+5 a y^{\prime }-6 y^{2}+6 a^{2} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \operatorname {WeierstrassP}\left (\frac {-c_{1} a +{\mathrm e}^{-a x}}{a}, 0, c_{2}\right ) {\mathrm e}^{-2 a x} \] |
Problem 9936
| ODE |
\[ \boxed {y^{\prime \prime }+3 a y^{\prime }-2 y^{3}+2 a^{2} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -\operatorname {JacobiSN}\left (\frac {\left (-c_{1} a +\sqrt {-{\mathrm e}^{-2 a x}}\right ) c_{2}}{a}, i\right ) c_{2} {\mathrm e}^{-a x} \] |
Problem 9937
| ODE |
\[ \boxed {y^{\prime \prime }-\frac {\left (3 n +4\right ) y^{\prime }}{n}-\frac {2 \left (1+n \right ) \left (2+n \right ) y \left (y^{\frac {n}{1+n}}-1\right )}{n^{2}}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9938
| ODE |
\[ \boxed {y^{\prime \prime }+a y^{\prime }+b y^{n}+\frac {\left (a^{2}-1\right ) y}{4}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9939
| ODE |
\[ \boxed {y^{\prime \prime }+a y^{\prime }+b \,x^{v} y^{n}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9940
| ODE |
\[ \boxed {y^{\prime \prime }+a y^{\prime }+b \,{\mathrm e}^{y}=2 a} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9941
| ODE |
\[ \boxed {y^{\prime \prime }+a y^{\prime }+f \left (x \right ) \sin \left (y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9942
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } y-y^{3}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {2 \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}}{\textit {\_a}^{4}-\textit {\_a}^{2} \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}+\left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {4 \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}}{i \textit {\_a}^{4} \sqrt {3}+\textit {\_a}^{4}-i \sqrt {3}\, \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}+2 \textit {\_a}^{2} \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}+\left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}}d \textit {\_a} = c_{3} +x \] Verified OK. \[ \int _{}^{y}\frac {4 \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}}{i \textit {\_a}^{4} \sqrt {3}-\textit {\_a}^{4}-i \sqrt {3}\, \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}-2 \textit {\_a}^{2} \left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {1}{3}}-\left (2 \,{\mathrm e}^{6 c_{1}}+\textit {\_a}^{6}+2 \sqrt {{\mathrm e}^{12 c_{1}}+{\mathrm e}^{6 c_{1}} \textit {\_a}^{6}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{4} \] Verified OK.
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Maple solution |
\begin{align*} 2 \left (\int _{}^{y \left (x \right )}\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}}{\textit {\_a}^{4}-\textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -4 \left (\int _{}^{y \left (x \right )}\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}}{-i \sqrt {3}\, \textit {\_a}^{4}+i \sqrt {3}\, \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 4 \left (\int _{}^{y \left (x \right )}-\frac {\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \textit {\_a}^{4}-i \sqrt {3}\, \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}+\textit {\_a}^{4}+2 \textit {\_a}^{2} \left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {1}{3}}+\left (\textit {\_a}^{6}+2 c_{1} +2 \sqrt {c_{1} \left (\textit {\_a}^{6}+c_{1} \right )}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9943
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } y-y^{3}+a y=0} \] |
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program solution |
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Maple solution | \[ -\frac {\left (\int _{}^{y \left (x \right )}\frac {4 {\operatorname {RootOf}\left (\left (-4 \textit {\_a}^{6}+12 \textit {\_a}^{4} a -12 \textit {\_a}^{2} a^{2}+4 a^{3}+320 c_{1} \right ) \textit {\_Z}^{9}+\left (-189 \textit {\_a}^{6}+567 \textit {\_a}^{4} a -567 \textit {\_a}^{2} a^{2}+189 a^{3}+15120 c_{1} \right ) \textit {\_Z}^{6}+238140 c_{1} \textit {\_Z}^{3}+1250235 c_{1} \right )}^{3}+63}{\textit {\_a}^{2}-a}d \textit {\_a} \right )}{63}-x -c_{2} = 0 \] |
Problem 9944
| ODE |
\[ \boxed {y^{\prime \prime }+\left (y+3 a \right ) y^{\prime }-y^{3}+a y^{2}+2 a^{2} y=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}-\frac {-\textit {\_f}^{8}+c_{1} \textit {\_f}^{2}-{\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1} \right ) {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {1}{3}}}d \textit {\_f} \right ) a +c_{2} a +{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}+\textit {\_f}^{8}+i \sqrt {3}\, c_{1} \textit {\_f}^{2}+i \sqrt {3}\, {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}-c_{1} \textit {\_f}^{2}+{\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1} \right ) {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {1}{3}}}d \textit {\_f} \right ) a +2 c_{2} a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ y \left (x \right ) &= \operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}-\textit {\_f}^{8}+i \sqrt {3}\, c_{1} \textit {\_f}^{2}+i \sqrt {3}\, {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}+c_{1} \textit {\_f}^{2}-{\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+c_{1} \right ) {\left (\left (-\textit {\_f}^{6}+c_{1} \right )^{2} \left (\sqrt {\frac {c_{1}}{-\textit {\_f}^{6}+c_{1}}}-1\right )\right )}^{\frac {1}{3}}}d \textit {\_f} \right ) a +2 c_{2} a +2 \,{\mathrm e}^{-a x}\right ) {\mathrm e}^{-a x} \\ \end{align*} |
Problem 9945
| ODE |
\[ \boxed {y^{\prime \prime }+\left (y+3 f \left (x \right )\right ) y^{\prime }-y^{3}+y^{2} f \left (x \right )+y \left (f^{\prime }\left (x \right )+2 f \left (x \right )^{2}\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9946
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime } y-y^{3}-\left (\frac {f^{\prime }\left (x \right )}{f \left (x \right )}+f \left (x \right )\right ) \left (3 y^{\prime }+y^{2}\right )+\left (f \left (x \right )^{2} a +3 f^{\prime }\left (x \right )+\frac {3 {f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}-\frac {f^{\prime \prime }\left (x \right )}{f \left (x \right )}\right ) y=-b f \left (x \right )^{3}} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9947
| ODE |
\[ \boxed {y^{\prime \prime }+\left (y-\frac {3 f^{\prime }\left (x \right )}{2 f \left (x \right )}\right ) y^{\prime }-y^{3}-\frac {f^{\prime }\left (x \right ) y^{2}}{2 f \left (x \right )}+\frac {\left (f \left (x \right )+\frac {{f^{\prime }\left (x \right )}^{2}}{f \left (x \right )^{2}}-f^{\prime \prime }\left (x \right )\right ) y}{2 f \left (x \right )}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9948
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime } y+f \left (x \right ) y^{\prime }+y f^{\prime }\left (x \right )=0} \] |
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program solution |
\[ y = \frac {\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{-c_{1} \textit {\_Y} \left (x \right )+2 f \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{\operatorname {DESol}\left (\left \{-c_{1} \textit {\_Y} \left (x \right )+2 f \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 9949
| ODE |
\[ \boxed {y^{\prime \prime }+2 y^{\prime } y+f \left (x \right ) \left (y^{2}+y^{\prime }\right )=g \left (x \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9950
| ODE |
\[ \boxed {y^{\prime \prime }+3 y^{\prime } y+y^{3}+f \left (x \right ) y=g \left (x \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9951
| ODE |
\[ \boxed {y^{\prime \prime }+\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right )=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )}d x \right )+c_{2}}{c_{1} \left (\int \left (\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )}d x \right )d x \right )+c_{2} x +1} \] |
Problem 9952
| ODE |
\[ \boxed {y^{\prime \prime }-3 y^{\prime } y-3 a y^{2}-4 a^{2} y=b} \] |
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program solution |
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Maple solution | \begin{align*} -6 a^{2} \left (\int _{}^{y \left (x \right )}\frac {1}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+{\operatorname {RootOf}\left (\left (\operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +\operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right ) \sqrt {4 a^{4}-3 a b}+\left (2 \operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +2 \operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right ) a^{2}+3 \textit {\_a} \left (\operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +\operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right ) a +\textit {\_Z} \operatorname {BesselK}\left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, -\frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} -\textit {\_Z} \operatorname {BesselI}\left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, -\frac {\textit {\_Z}}{2 a^{2}}\right )\right )}^{2}-3 a b}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -6 a^{2} \left (\int _{}^{y \left (x \right )}\frac {1}{-12 \textit {\_a} \,a^{3}-9 \textit {\_a}^{2} a^{2}+{\operatorname {RootOf}\left (\left (\operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +\operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right ) \sqrt {4 a^{4}-3 a b}+\left (2 \operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +2 \operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right ) a^{2}+3 \textit {\_a} \left (\operatorname {BesselK}\left (\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} +\operatorname {BesselI}\left (-\frac {4 a^{3}-3 b}{2 a \sqrt {4 a^{4}-3 a b}}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right ) a +\textit {\_Z} \operatorname {BesselK}\left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, \frac {\textit {\_Z}}{2 a^{2}}\right ) c_{1} -\textit {\_Z} \operatorname {BesselI}\left (\frac {-4 a^{3}+2 \sqrt {4 a^{4}-3 a b}\, a +3 b}{2 \sqrt {4 a^{4}-3 a b}\, a}, \frac {\textit {\_Z}}{2 a^{2}}\right )\right )}^{2}-3 a b}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9953
| ODE |
\[ \boxed {y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right )=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {-c_{1} \left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right )-c_{2}}{c_{1} \left (\int \int {\mathrm e}^{\int f \left (x \right )d x}d x d x \right )+c_{2} x +1} \] |
Problem 9954
| ODE |
\[ \boxed {y^{\prime \prime }-2 a y^{\prime } y=0} \] |
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program solution |
\[ y = \frac {\tan \left (c_{2} \sqrt {c_{1} a}+x \sqrt {c_{1} a}\right ) \sqrt {c_{1} a}}{a} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\tan \left (\left (c_{2} +x \right ) \sqrt {c_{1} a}\right ) \sqrt {c_{1} a}}{a} \] |
Problem 9955
| ODE |
\[ \boxed {y^{\prime \prime }+a y^{\prime } y+b y^{3}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (b \,\textit {\_a}^{4}+a \,\textit {\_a}^{2} \textit {\_Z} +2 \textit {\_Z}^{2}-{\mathrm e}^{\operatorname {RootOf}\left ({\tanh \left (\frac {\sqrt {a^{2}-8 b}\, \left (4 c_{1} -\textit {\_Z} \right )}{2 a}\right )}^{2} \textit {\_a}^{4} a^{2}-8 {\tanh \left (\frac {\sqrt {a^{2}-8 b}\, \left (4 c_{1} -\textit {\_Z} \right )}{2 a}\right )}^{2} \textit {\_a}^{4} b -a^{2} \textit {\_a}^{4}+8 b \,\textit {\_a}^{4}-8 \,{\mathrm e}^{\textit {\_Z}}\right )}\right )}d \textit {\_a} = x +c_{2} \] Warning, solution could not be verified |
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Maple solution | \[ \int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (-2 a \,\textit {\_a}^{2} \operatorname {arctanh}\left (\frac {a \,\textit {\_a}^{2}+4 \textit {\_Z}}{\sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}}\right )-\ln \left (\textit {\_a}^{4} b +\textit {\_Z} \,\textit {\_a}^{2} a +2 \textit {\_Z}^{2}\right ) \sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}+c_{1} \sqrt {\textit {\_a}^{4} \left (a^{2}-8 b \right )}\right )}d \textit {\_a} -x -c_{2} = 0 \] |
Problem 9956
| ODE |
\[ \boxed {y^{\prime \prime }+f \left (x , y\right ) y^{\prime }+g \left (x , y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9957
| ODE |
\[ \boxed {y^{\prime \prime }+a {y^{\prime }}^{2}+y b=0} \] |
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program solution |
\[ \int _{}^{y}\frac {2 \,{\mathrm e}^{2 \textit {\_a} a} a}{\sqrt {-2 \,{\mathrm e}^{2 \textit {\_a} a} \left (2 \,{\mathrm e}^{2 \textit {\_a} a} a b \textit {\_a} -4 c_{1} a^{2}-{\mathrm e}^{2 \textit {\_a} a} b \right )}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {2 \,{\mathrm e}^{2 \textit {\_a} a} a}{\sqrt {-2 \,{\mathrm e}^{2 \textit {\_a} a} \left (2 \,{\mathrm e}^{2 \textit {\_a} a} a b \textit {\_a} -4 c_{1} a^{2}-{\mathrm e}^{2 \textit {\_a} a} b \right )}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} -2 a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \textit {\_a} b a +2 b}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ 2 a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \textit {\_a} b a +2 b}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9958
| ODE |
\[ \boxed {y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b y^{\prime }+y c=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9959
| ODE |
\[ \boxed {y^{\prime \prime }+a {y^{\prime }}^{2}+b y^{\prime }+y c=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9960
| ODE |
\[ \boxed {y^{\prime \prime }+a {y^{\prime }}^{2}+b \sin \left (y\right )=0} \] |
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program solution |
\[ \int _{}^{y}\frac {\sqrt {2}\, {\mathrm e}^{2 \textit {\_a} a} \left (4 a^{2}+1\right )}{2 \sqrt {{\mathrm e}^{2 \textit {\_a} a} \left (4 a^{2}+1\right ) \left (-2 \sin \left (\textit {\_a} \right ) {\mathrm e}^{2 \textit {\_a} a} a b +\cos \left (\textit {\_a} \right ) {\mathrm e}^{2 \textit {\_a} a} b +4 c_{1} a^{2}+c_{1} \right )}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {\sqrt {2}\, {\mathrm e}^{2 \textit {\_a} a} \left (4 a^{2}+1\right )}{2 \sqrt {{\mathrm e}^{2 \textit {\_a} a} \left (4 a^{2}+1\right ) \left (-2 \sin \left (\textit {\_a} \right ) {\mathrm e}^{2 \textit {\_a} a} a b +\cos \left (\textit {\_a} \right ) {\mathrm e}^{2 \textit {\_a} a} b +4 c_{1} a^{2}+c_{1} \right )}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} 4 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_{1} +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} \right ) a^{2}+\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_{1} +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} -c_{2} -x &= 0 \\ -4 \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_{1} +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} \right ) a^{2}-\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (4 a^{2}+1\right ) \left (4 \,{\mathrm e}^{-2 a \textit {\_a}} c_{1} a^{2}-4 \sin \left (\textit {\_a} \right ) a b +{\mathrm e}^{-2 a \textit {\_a}} c_{1} +2 \cos \left (\textit {\_a} \right ) b \right )}}d \textit {\_a} \right )-c_{2} -x &= 0 \\ \end{align*} |
Problem 9961
| ODE |
\[ \boxed {y^{\prime \prime }+a y^{\prime } {| y^{\prime }|}+b \sin \left (y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9962
| ODE |
\[ \boxed {y^{\prime \prime }+a y {y^{\prime }}^{2}+y b=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (-\left (\textit {\_Z}^{2} a +b \right )^{\frac {1}{2 a}}+c_{2} {\mathrm e}^{-\frac {\textit {\_a}^{2}}{2}+c_{1}}\right )}d \textit {\_a} = c_{3} +x \] Warning, solution could not be verified |
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Maple solution | \begin{align*} a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \left ({\mathrm e}^{-a \,\textit {\_a}^{2}} c_{1} a -b \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \left ({\mathrm e}^{-a \,\textit {\_a}^{2}} c_{1} a -b \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9963
| ODE |
\[ \boxed {y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (x \right ) y^{\prime }=0} \] |
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program solution |
\[ \int _{0}^{y}{\mathrm e}^{\int f \left (y \right )d \textit {\_a}}d \textit {\_a} = c_{1} \left (\int {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}d x \right )+c_{2} \] Verified OK.
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Maple solution | \[ \int _{}^{y \left (x \right )}{\mathrm e}^{\int f \left (\textit {\_b} \right )d \textit {\_b}}d \textit {\_b} -c_{1} \left (\int {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}d x \right )-c_{2} = 0 \] |
Problem 9964
| ODE |
\[ \boxed {y^{\prime \prime }-\frac {D\left (f \right )\left (y\right ) {y^{\prime }}^{3}}{f \left (y\right )}+g \left (x \right ) y^{\prime }+h \left (x \right ) f \left (y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9965
| ODE |
\[ \boxed {y^{\prime \prime }+\phi \left (y\right ) {y^{\prime }}^{2}+f \left (x \right ) y^{\prime }+g \left (x \right ) \Phi \left (y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9966
| ODE |
\[ \boxed {y^{\prime \prime }+f \left (y\right ) {y^{\prime }}^{2}+g \left (y\right ) y^{\prime }+h \left (y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9967
| ODE |
\[ \boxed {y^{\prime \prime }+\left ({y^{\prime }}^{2}+1\right ) \left (f \left (x , y\right ) y^{\prime }+g \left (x , y\right )\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9968
| ODE |
\[ \boxed {y^{\prime \prime }+a y \left ({y^{\prime }}^{2}+1\right )^{2}=0} \] |
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program solution |
\[ \int _{}^{y}-\frac {\textit {\_a}^{2} a -2 c_{1}}{\sqrt {-\left (\textit {\_a}^{2} a -2 c_{1} \right ) \left (\textit {\_a}^{2} a -2 c_{1} -1\right )}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}\frac {\textit {\_a}^{2} a -2 c_{1}}{\sqrt {-\left (\textit {\_a}^{2} a -2 c_{1} \right ) \left (\textit {\_a}^{2} a -2 c_{1} -1\right )}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}+2 c_{1}}{\sqrt {-\left (-1+a \left (\textit {\_a}^{2}+2 c_{1} \right )\right ) \left (\textit {\_a}^{2}+2 c_{1} \right ) a}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}+2 c_{1}}{\sqrt {-\left (-1+a \left (\textit {\_a}^{2}+2 c_{1} \right )\right ) \left (\textit {\_a}^{2}+2 c_{1} \right ) a}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9969
| ODE |
\[ \boxed {y^{\prime \prime }-a \left (y^{\prime } x -y\right )^{v}=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (2^{\frac {1}{v -1}} \left (\int -\frac {\left (\left (v -1\right ) a \,x^{2}-c_{1} \right ) \left (-\frac {1}{\left (v -1\right ) a \,x^{2}-c_{1}}\right )^{\frac {v}{v -1}}}{x^{2}}d x \right )+c_{2} \right ) x \] |
Problem 9970
| ODE |
\[ \boxed {y^{\prime \prime }-k \,x^{a} y^{b} {y^{\prime }}^{r}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9971
| ODE |
\[ \boxed {y^{\prime \prime }+\left (y^{\prime }-\frac {y}{x}\right )^{a} f \left (x , y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9972
| ODE |
\[ \boxed {y^{\prime \prime }-a \sqrt {{y^{\prime }}^{2}+1}=0} \] |
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program solution |
\[ y = \frac {\left (-2 c_{1} a^{2} {\mathrm e}^{c_{2} \sqrt {a^{2}}+\sqrt {a^{2}}\, x}+{\mathrm e}^{2 c_{2} \sqrt {a^{2}}+2 \sqrt {a^{2}}\, x} \sqrt {a^{2}}+\sqrt {a^{2}}\right ) {\mathrm e}^{-c_{2} \sqrt {a^{2}}-\sqrt {a^{2}}\, x}}{2 a^{2}} \] Verified OK. \[ y = \frac {\left (-2 c_{1} a^{2} {\mathrm e}^{-c_{3} \sqrt {a^{2}}-\sqrt {a^{2}}\, x}+{\mathrm e}^{-2 c_{3} \sqrt {a^{2}}-2 \sqrt {a^{2}}\, x} \sqrt {a^{2}}+\sqrt {a^{2}}\right ) {\mathrm e}^{c_{3} \sqrt {a^{2}}+\sqrt {a^{2}}\, x}}{2 a^{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= c_{2} +\frac {\cosh \left (a \left (c_{1} +x \right )\right )}{a} \\ \end{align*} |
Problem 9973
| ODE |
\[ \boxed {y^{\prime \prime }-a \sqrt {{y^{\prime }}^{2}+1}=b} \] |
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program solution |
\[ -\frac {b \left (\frac {\ln \left (\frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}+b^{2}}{\sqrt {b^{2}}}+\sqrt {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )^{2} b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}-a^{2}+b^{2}}\right )}{\sqrt {b^{2}}}-\frac {\ln \left (\frac {-2 a^{2}+2 b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}+2 \sqrt {-a^{2}+b^{2}}\, \sqrt {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )^{2} b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}-a^{2}+b^{2}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )}\right )}{\sqrt {-a^{2}+b^{2}}}\right )}{a} = x +c_{2} \] Warning, solution could not be verified \[ \frac {b \left (\frac {\ln \left (\frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}+b^{2}}{\sqrt {b^{2}}}+\sqrt {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )^{2} b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}-a^{2}+b^{2}}\right )}{\sqrt {b^{2}}}-\frac {\ln \left (\frac {-2 a^{2}+2 b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}+2 \sqrt {-a^{2}+b^{2}}\, \sqrt {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )^{2} b^{2}+2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right ) b^{2}-a^{2}+b^{2}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {a^{2} y+c_{1} a^{2}+b}{b}}}{b}\right )}\right )}{\sqrt {-a^{2}+b^{2}}}\right )}{a} = c_{3} +x \] Warning, solution could not be verified |
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Maple solution | \[ y \left (x \right ) = \int \operatorname {RootOf}\left (x -\left (\int _{}^{\textit {\_Z}}\frac {1}{a \sqrt {\textit {\_f}^{2}+1}+b}d \textit {\_f} \right )+c_{1} \right )d x +c_{2} \] |
Problem 9974
| ODE |
\[ \boxed {y^{\prime \prime }-a \sqrt {{y^{\prime }}^{2}+y^{2} b}=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (x -\left (\int _{}^{\textit {\_Z}}-\frac {1}{\textit {\_f}^{2}-a \sqrt {\textit {\_f}^{2}+b}}d \textit {\_f} \right )+c_{1} \right )d x +c_{2}} \\ \end{align*} |
Problem 9975
| ODE |
\[ \boxed {y^{\prime \prime }-a \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \] |
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program solution |
\[ -\frac {\sqrt {1-\left (y+c_{1} \right )^{2} a^{2}}}{a} = x +c_{2} \] Verified OK. \[ \frac {\sqrt {1-\left (y+c_{1} \right )^{2} a^{2}}}{a} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= \frac {\left (-1+\left (c_{1} +x \right )^{2} a^{2}\right ) \sqrt {-\frac {1}{-1+\left (c_{1} +x \right )^{2} a^{2}}}+c_{2} a}{a} \\ \end{align*} |
Problem 9976
| ODE |
\[ \boxed {y^{\prime \prime }-2 a x \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \] |
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program solution |
\[ y = \frac {c_{1} \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )}{\sqrt {-\frac {a}{c_{1} +1}}}-\frac {\left (\operatorname {EllipticF}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {a}{c_{1} +1}}, \sqrt {\frac {c_{1} +1}{c_{1} -1}}\right )\right ) \sqrt {\frac {x^{2} a +c_{1} -1}{c_{1} -1}}\, \sqrt {\frac {x^{2} a +c_{1} +1}{c_{1} +1}}\, \left (c_{1} -1\right ) \sqrt {-\frac {1}{a^{2} x^{4}+2 a c_{1} x^{2}+c_{1}^{2}-1}}}{\sqrt {-\frac {a}{c_{1} +1}}}+c_{2} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= a \left (\int \sqrt {-\frac {1}{-1+\left (x^{2}+2 c_{1} \right )^{2} a^{2}}}\, \left (x^{2}+2 c_{1} \right )d x \right )+c_{2} \\ \end{align*} |
Problem 9977
| ODE |
\[ \boxed {y^{\prime \prime }-a y \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_a}^{2} a c_{1} -4 c_{1}^{2}+4}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {\textit {\_a}^{2} a +2 c_{1}}{\sqrt {-\textit {\_a}^{4} a^{2}-4 \textit {\_a}^{2} a c_{1} -4 c_{1}^{2}+4}}d \textit {\_a} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}+2 c_{1}}{\sqrt {4-\left (\textit {\_a}^{2}+2 c_{1} \right )^{2} a^{2}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -a \left (\int _{}^{y \left (x \right )}\frac {\textit {\_a}^{2}+2 c_{1}}{\sqrt {4-\left (\textit {\_a}^{2}+2 c_{1} \right )^{2} a^{2}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*} |
Problem 9978
| ODE |
\[ \boxed {y^{\prime \prime }-2 a \left (c +b x +y\right ) \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= -b x +\operatorname {RootOf}\left (c_{2} b^{3}-b^{3} x +c_{2} b -b x +\int _{}^{\textit {\_Z}}\frac {4 b^{2} a^{2} \textit {\_f}^{2} c^{2}+4 c \,b^{2} a^{2} \textit {\_f}^{3}+a^{2} b^{2} \textit {\_f}^{4}-8 b^{2} c_{1} a^{2} c \textit {\_f} -4 c_{1} a^{2} b^{2} \textit {\_f}^{2}+4 c_{1}^{2} a^{2} b^{2}-2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1} -8 \textit {\_f} \,a^{2} c c_{1} +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a c \textit {\_f} -\sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1} -8 \textit {\_f} \,a^{2} c c_{1} +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a \,\textit {\_f}^{2}-b^{4}+2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1} -8 \textit {\_f} \,a^{2} c c_{1} +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, c_{1} a -b^{2}}{a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1} -8 \textit {\_f} \,a^{2} c c_{1} +4 a^{2} c_{1}^{2}-b^{2}-1}d \textit {\_f} \right ) \\ y \left (x \right ) &= -b x +\operatorname {RootOf}\left (c_{2} b^{3}-b^{3} x +c_{2} b -b x -\left (\int _{}^{\textit {\_Z}}-\frac {4 b^{2} a^{2} \textit {\_f}^{2} c^{2}+4 c \,b^{2} a^{2} \textit {\_f}^{3}+a^{2} b^{2} \textit {\_f}^{4}-8 b^{2} c_{1} a^{2} c \textit {\_f} -4 c_{1} a^{2} b^{2} \textit {\_f}^{2}+4 c_{1}^{2} a^{2} b^{2}+2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1} -8 \textit {\_f} \,a^{2} c c_{1} +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a c \textit {\_f} +\sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1} -8 \textit {\_f} \,a^{2} c c_{1} +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, a \,\textit {\_f}^{2}-b^{4}-2 \sqrt {-b^{2} \left (a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1} -8 \textit {\_f} \,a^{2} c c_{1} +4 a^{2} c_{1}^{2}-b^{2}-1\right )}\, c_{1} a -b^{2}}{a^{2} \textit {\_f}^{4}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1} -8 \textit {\_f} \,a^{2} c c_{1} +4 a^{2} c_{1}^{2}-b^{2}-1}d \textit {\_f} \right )\right ) \\ \end{align*} |
Problem 9979
| ODE |
\[ \boxed {y^{\prime \prime }+y^{3} y^{\prime }-y y^{\prime } \sqrt {y^{4}+4 y^{\prime }}=0} \] |
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program solution |
\[ y = \tan \left (c_{2} \sqrt {{\mathrm e}^{4 c_{1}} {\mathrm e}^{2 c_{1}}}+x \sqrt {{\mathrm e}^{4 c_{1}} {\mathrm e}^{2 c_{1}}}\right ) \sqrt {{\mathrm e}^{4 c_{1}} {\mathrm e}^{2 c_{1}}}\, {\mathrm e}^{-2 c_{1}} \] Verified OK. \[ y = \frac {{\mathrm e}^{c_{1}} c_{3}^{2} {\mathrm e}^{2 x \,{\mathrm e}^{3 c_{1}}}-{\mathrm e}^{c_{1}}}{1+c_{3}^{2} {\mathrm e}^{2 x \,{\mathrm e}^{3 c_{1}}}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}}}{4 c_{1} +3 x} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{8 c_{1} +6 x} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (\left (4 c_{1} +3 x \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{8 c_{1} +6 x} \\ y \left (x \right ) &= \tan \left (\frac {c_{2} +x}{c_{1}^{3}}\right ) \sqrt {\frac {1}{c_{1}^{2}}} \\ y \left (x \right ) &= \tanh \left (\frac {c_{2} +x}{c_{1}^{3}}\right ) \sqrt {\frac {1}{c_{1}^{2}}} \\ \end{align*} |
Problem 9980
| ODE |
\[ \boxed {y^{\prime \prime }-f \left (y^{\prime }, x a +y b \right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9981
| ODE |
\[ \boxed {y^{\prime \prime }-y f \left (x , \frac {y^{\prime }}{y}\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9982
| ODE |
\[ \boxed {y^{\prime \prime }-x^{-2+n} f \left (y x^{-n}, y^{\prime } x^{1-n}\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9983
| ODE |
\[ \boxed {8 y^{\prime \prime }+9 {y^{\prime }}^{4}=0} \] |
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program solution |
\[ -\left (y+c_{1} \right )^{\frac {3}{2}} = x +c_{2} \] Verified OK. \[ \left (y+c_{1} \right )^{\frac {3}{2}} = c_{3} +x \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \left (c_{1} +x \right )^{\frac {2}{3}}+c_{2} \\ y \left (x \right ) &= -\frac {i \left (c_{1} +x \right )^{\frac {2}{3}} \sqrt {3}}{2}-\frac {\left (c_{1} +x \right )^{\frac {2}{3}}}{2}+c_{2} \\ y \left (x \right ) &= \frac {i \left (c_{1} +x \right )^{\frac {2}{3}} \sqrt {3}}{2}-\frac {\left (c_{1} +x \right )^{\frac {2}{3}}}{2}+c_{2} \\ \end{align*} |
Problem 9984
| ODE |
\[ \boxed {a y^{\prime \prime }+h \left (y^{\prime }\right )+y c=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9985
| ODE |
\[ \boxed {y^{\prime \prime } x +2 y^{\prime }-x y^{n}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9986
| ODE |
\[ \boxed {y^{\prime \prime } x +2 y^{\prime }+a \,x^{v} y^{n}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9987
| ODE |
\[ \boxed {y^{\prime \prime } x +2 y^{\prime }+x \,{\mathrm e}^{y}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9988
| ODE |
\[ \boxed {y^{\prime \prime } x +a y^{\prime }+b x \,{\mathrm e}^{y}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9989
| ODE |
\[ \boxed {y^{\prime \prime } x +a y^{\prime }+b \,x^{-2 a +5} {\mathrm e}^{y}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9990
| ODE |
\[ \boxed {y^{\prime \prime } x +\left (y-1\right ) y^{\prime }=0} \] |
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program solution |
\[ -\frac {2 \,\operatorname {arctanh}\left (\frac {-2 y+4}{2 \sqrt {2 c_{1} +4}}\right )}{\sqrt {2 c_{1} +4}}-\ln \left (x \right )-c_{2} = 0 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 c_{1} +\tanh \left (\frac {\ln \left (x \right )-c_{2}}{2 c_{1}}\right )}{c_{1}} \] |
Problem 9991
| ODE |
\[ \boxed {y^{\prime \prime } x -x^{2} {y^{\prime }}^{2}+2 y^{\prime }+y^{2}=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{2} -\left (\int _{}^{\textit {\_Z}}\frac {1}{-2 \textit {\_f} -1+{\mathrm e}^{\textit {\_f}} c_{1}}d \textit {\_f} \right )\right )}{x} \] |
Problem 9992
| ODE |
\[ \boxed {y^{\prime \prime } x +a \left (y^{\prime } x -y\right )^{2}=b} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\left (i \sqrt {b}\, \left (\int \frac {\tan \left (-i \sqrt {a}\, \sqrt {b}\, x +c_{1} \right )}{x^{2}}d x \right )+c_{2} \sqrt {a}\right ) x}{\sqrt {a}} \] |
Problem 9993
| ODE |
\[ \boxed {2 y^{\prime \prime } x +{y^{\prime }}^{3}+y^{\prime }=0} \] |
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program solution |
\[ y = -2 \sqrt {-c_{2}^{2}+x}\, c_{2} +c_{3} \] Verified OK. \[ y = 2 \sqrt {-c_{2}^{2}+x}\, c_{2} +c_{4} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {c_{2} c_{1} +2 \sqrt {c_{1} x -1}}{c_{1}} \\ y \left (x \right ) &= \frac {c_{2} c_{1} -2 \sqrt {c_{1} x -1}}{c_{1}} \\ \end{align*} |
Problem 9994
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-a \left (y^{n}-y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9995
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+a \left ({\mathrm e}^{y}-1\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9996
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-\left (2 a +b -1\right ) x y^{\prime }+\left (c^{2} b^{2} x^{2 b}+a \left (a +b \right )\right ) y=0} \] |
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program solution |
\[ y = \frac {c_{1} x^{a +\frac {b}{2}} \sqrt {2}\, \cos \left (c \,x^{b}\right )}{\sqrt {\pi }\, \sqrt {c \,x^{b}}}+\frac {c_{2} x^{a +\frac {b}{2}} \sqrt {2}\, \sin \left (c \,x^{b}\right )}{\sqrt {\pi }\, \sqrt {c \,x^{b}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{a} \left (c_{1} \sin \left (x^{b} c \right )+c_{2} \cos \left (x^{b} c \right )\right ) \] |
Problem 9997
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+\left (1+a \right ) x y^{\prime }-x^{k} f \left (x^{k} y, y^{\prime } x +k y\right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 9998
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+a \left (y^{\prime } x -y\right )^{2}=b \,x^{2}} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\left (-\sqrt {-a b}\, \left (\int \frac {\operatorname {BesselY}\left (1, \sqrt {-a b}\, x \right ) c_{1} +\operatorname {BesselJ}\left (1, \sqrt {-a b}\, x \right )}{x \left (c_{1} \operatorname {BesselY}\left (0, \sqrt {-a b}\, x \right )+\operatorname {BesselJ}\left (0, \sqrt {-a b}\, x \right )\right )}d x \right )+c_{2} a \right ) x}{a} \] |
Problem 9999
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+a y {y^{\prime }}^{2}=-b x} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 10000
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-\sqrt {a \,x^{2} {y^{\prime }}^{2}+y^{2} b}=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right )-{\mathrm e}^{\int _{}^{\ln \left (x \right )}\operatorname {RootOf}\left (-y \left (x \right ) \left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{2} y \left (x \right )-\textit {\_a} y \left (x \right )-\sqrt {y \left (x \right )^{2} \left (a \,\textit {\_a}^{2}+b \right )}}d \textit {\_a} \right )-\textit {\_b} +c_{1} \right )d \textit {\_b} +c_{2}} &= 0 \\ \end{align*} |