Problem 11101
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a +b \right ) {\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (b \,{\mathrm e}^{\lambda x}+\lambda \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {a \,{\mathrm e}^{x \lambda }}{\lambda }} \left (c_{1} +\operatorname {expIntegral}_{1}\left (-\frac {{\mathrm e}^{x \lambda } \left (a -b \right )}{\lambda }\right ) c_{2} \right ) \] |
Problem 11102
| ODE |
\[ \boxed {y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+{\mathrm e}^{\mu x} b +\mu \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\left (\int {\mathrm e}^{\frac {-2 b \,{\mathrm e}^{x \mu } \lambda -{\mathrm e}^{x \lambda } a \mu }{\mu \lambda }}d x \right ) c_{1} +c_{2} \right ) {\mathrm e}^{\frac {b \,{\mathrm e}^{x \mu }}{\mu }} \] |
Problem 11103
| ODE |
\[ \boxed {y^{\prime \prime }+2 k \,{\mathrm e}^{\mu x} y^{\prime }+\left (a \,{\mathrm e}^{2 \lambda x}+b \,{\mathrm e}^{\lambda x}+k^{2} {\mathrm e}^{2 \mu x}+k \mu \,{\mathrm e}^{\mu x}+c \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 11104
| ODE |
\[ \boxed {y^{\prime \prime }-\left (a +2 \,{\mathrm e}^{a x} b \right ) y^{\prime }+b^{2} {\mathrm e}^{2 a x} y=0} \] |
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program solution |
\[ y = \left (c_{1} {\mathrm e}^{\frac {\sqrt {a^{2}}\, x}{2}}+c_{2} {\mathrm e}^{-\frac {\sqrt {a^{2}}\, x}{2}}\right ) {\mathrm e}^{\frac {a^{2} x +2 \,{\mathrm e}^{a x} b}{2 a}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{\frac {a^{2} x +2 \,{\mathrm e}^{a x} b}{2 a}} \left (c_{1} \sinh \left (\frac {a x}{2}\right )+c_{2} \cosh \left (\frac {a x}{2}\right )\right ) \] |
Problem 11105
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{2} \sqrt {\pi }\, \left ({\mathrm e}^{x \lambda } a +{\mathrm e}^{-x \lambda } \lambda \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, {\mathrm e}^{x \lambda } \sqrt {a}}{2 \sqrt {\lambda }}\right )+\sqrt {a}\, \sqrt {\lambda }\, {\mathrm e}^{-\frac {a \,{\mathrm e}^{2 x \lambda }}{2 \lambda }} \sqrt {2}\, c_{2} +c_{1} \left ({\mathrm e}^{x \lambda } a +{\mathrm e}^{-x \lambda } \lambda \right ) \] |
Problem 11106
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+y b \,{\mathrm e}^{2 \lambda x}=0} \] |
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program solution |
\[ y = c_{1} \left (-\frac {{\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}}{a}\right )^{-\frac {-a +\sqrt {a^{2}-4 b}}{2 a}}+c_{2} \left (-\frac {{\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}}{a}\right )^{\frac {a +\sqrt {a^{2}-4 b}}{2 a}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{x \lambda }}{2 \lambda }}+c_{2} {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{x \lambda }}{2 \lambda }} \] |
Problem 11107
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a -\left (b +3 \lambda \right ) x \lambda }{2 \lambda }} c_{2} \left (-\lambda -2 c +b \right )^{2} \operatorname {WhittakerM}\left (-\frac {-\lambda -2 c +b}{2 \lambda }, -\frac {-2 \lambda -2 c +b}{2 \lambda }, \frac {a \,{\mathrm e}^{x \lambda }}{\lambda }\right )+\left (\left (\lambda +2 c -b \right ) {\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a -\left (b +3 \lambda \right ) x \lambda }{2 \lambda }}+a \,{\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a -x \lambda \left (b +\lambda \right )}{2 \lambda }}\right ) c_{2} \lambda \operatorname {WhittakerM}\left (-\frac {b -2 c +\lambda }{2 \lambda }, -\frac {-2 \lambda -2 c +b}{2 \lambda }, \frac {a \,{\mathrm e}^{x \lambda }}{\lambda }\right )+c_{1} {\mathrm e}^{-c x} \] |
Problem 11108
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a +b \,{\mathrm e}^{2 \lambda x}\right ) y^{\prime }+\lambda \left (a -\lambda -b \,{\mathrm e}^{2 \lambda x}\right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 11109
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a +b \,{\mathrm e}^{\lambda x}+b -3 \lambda \right ) y^{\prime }+a^{2} \lambda \left (b -\lambda \right ) {\mathrm e}^{2 \lambda x} y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {{\mathrm e}^{x \lambda } \left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right )}{2 \lambda }} \left (\operatorname {KummerU}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (-2 \lambda +b +a \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {-2 \lambda +b +a}{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{2} +\operatorname {KummerM}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (-2 \lambda +b +a \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {-2 \lambda +b +a}{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} \right ) \] |
Problem 11110
| ODE |
\[ \boxed {y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+{\mathrm e}^{\mu x} c \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 11111
| ODE |
\[ \boxed {y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a \left (b +\lambda \right ) {\mathrm e}^{\lambda x}+c \right ) y=0} \] |
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program solution |
\[ y = \left (c_{1} {\mathrm e}^{-\frac {\sqrt {b^{2}-4 c}\, x}{2}}+c_{2} {\mathrm e}^{\frac {\sqrt {b^{2}-4 c}\, x}{2}}\right ) {\mathrm e}^{-\frac {\lambda b x +2 a \,{\mathrm e}^{\lambda x}}{2 \lambda }} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {b \lambda x +2 \,{\mathrm e}^{x \lambda } a}{2 \lambda }} \left (c_{1} \sinh \left (\frac {\sqrt {b^{2}-4 c}\, x}{2}\right )+c_{2} \cosh \left (\frac {\sqrt {b^{2}-4 c}\, x}{2}\right )\right ) \] |
Problem 11112
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left ({\mathrm e}^{2 \lambda x} c +a b \,{\mathrm e}^{\lambda x}+b^{2}-\lambda b \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {-2 b \lambda x +{\mathrm e}^{x \lambda } \sqrt {a^{2}-4 c}-{\mathrm e}^{x \lambda } a}{2 \lambda }}+c_{2} {\mathrm e}^{-\frac {2 b \lambda x +{\mathrm e}^{x \lambda } \sqrt {a^{2}-4 c}+{\mathrm e}^{x \lambda } a}{2 \lambda }} \] |
Problem 11113
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -\operatorname {WhittakerM}\left (-\frac {b}{2}+k , -\frac {b}{2}+k +\frac {1}{2}, \left (-2 c +a \right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{x}}{2}-\frac {b x}{2}} \left (-2 c +a \right )^{-b +2 k} c_{2} +\left (\left (-2 c +a \right ) {\mathrm e}^{x}\right )^{-\frac {b}{2}+k} c_{2} \left (-2 c +a \right )^{-b +2 k} \left (-1+b -2 k \right ) {\mathrm e}^{\left (-a +c \right ) {\mathrm e}^{x}-\frac {b x}{2}}+c_{1} {\mathrm e}^{-k x -{\mathrm e}^{x} c} \] |
Problem 11114
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a -x \lambda \left (b +\lambda \right )}{2 \lambda }} \left (\operatorname {WhittakerM}\left (-\frac {a \left (b +\lambda \right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {a \left (b +\lambda \right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{x \lambda }}{\lambda }\right ) c_{2} \right ) \] |
Problem 11115
| ODE |
\[ \boxed {y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+{\mathrm e}^{2 \mu x} b +{\mathrm e}^{\mu x} c +k \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 11116
| ODE |
\[ \boxed {y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b -\lambda \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a b \,{\mathrm e}^{\lambda x}+{\mathrm e}^{2 \mu x} c +d \,{\mathrm e}^{\mu x}+k \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 11117
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+{\mathrm e}^{\mu x} b \right ) y^{\prime }+a \,{\mathrm e}^{\lambda x} \left ({\mathrm e}^{\mu x} b +\lambda \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 11118
| ODE |
\[ \boxed {y^{\prime \prime }+{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{2 \mu x}+b \right ) y^{\prime }+\mu \left ({\mathrm e}^{\lambda x} \left (b -a \,{\mathrm e}^{2 \mu x}\right )-\mu \right ) y=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = \left (\left (\int \frac {{\mathrm e}^{\frac {-{\mathrm e}^{x \left (\lambda +2 \mu \right )} a \lambda -2 \left (\frac {\lambda }{2}+\mu \right ) \left (-2 \lambda \mu x +b \,{\mathrm e}^{x \lambda }\right )}{\lambda \left (\lambda +2 \mu \right )}}}{\left ({\mathrm e}^{2 x \mu } a +b \right )^{2}}d x \right ) c_{2} +c_{1} \right ) \left (a \,{\mathrm e}^{x \mu }+{\mathrm e}^{-x \mu } b \right ) \] |
Problem 11119
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+{\mathrm e}^{\mu x} b +c \right ) y^{\prime }+\left (a \lambda \,{\mathrm e}^{\lambda x}+\mu \,{\mathrm e}^{\mu x} b \right ) y=0} \] |
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program solution |
\[ y = {\mathrm e}^{\frac {-a \,{\mathrm e}^{\lambda x} \mu -\lambda \left (c x \mu +{\mathrm e}^{\mu x} b \right )}{\mu \lambda }} \left (c_{1} \left (\int {\mathrm e}^{\frac {c x \lambda \mu +a \,{\mathrm e}^{\lambda x} \mu +{\mathrm e}^{\mu x} b \lambda }{\lambda \mu }}d x \right )+c_{2} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{1} \left (\int {\mathrm e}^{\frac {c x \mu \lambda +{\mathrm e}^{x \lambda } a \mu +b \,{\mathrm e}^{x \mu } \lambda }{\mu \lambda }}d x \right )+c_{2} \right ) {\mathrm e}^{\frac {-{\mathrm e}^{x \lambda } a \mu -\lambda \left (c x \mu +b \,{\mathrm e}^{x \mu }\right )}{\mu \lambda }} \] |
Problem 11120
| ODE |
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+{\mathrm e}^{\mu x} b +c \right ) y^{\prime }+\left (a b \,{\mathrm e}^{x \left (\lambda +\mu \right )}+{\mathrm e}^{\lambda x} a c +\mu \,{\mathrm e}^{\mu x} b \right ) y=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 11121
| ODE |
\[ \boxed {\frac {2 x y+1}{y}+\frac {\left (y-x \right ) y^{\prime }}{y^{2}}=0} \] |
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program solution |
\[ y = {\mathrm e}^{-x^{2}+\operatorname {LambertW}\left (-x \,{\mathrm e}^{x^{2}-c_{1}}\right )+c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x}{\operatorname {LambertW}\left (-{\mathrm e}^{x^{2}} c_{1} x \right )} \] |
Problem 11122
| ODE |
\[ \boxed {\frac {y^{2}-2 x^{2}}{y^{2} x -x^{3}}+\frac {\left (2 y^{2}-x^{2}\right ) y^{\prime }}{y^{3}-x^{2} y}=0} \] |
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program solution |
\[ \frac {\ln \left (y+x \right )}{2}+\frac {\ln \left (x -y\right )}{2}+\ln \left (x \right )+\ln \left (y\right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {\frac {2 c_{1} x^{3}-2 \sqrt {c_{1}^{2} x^{6}+4}}{c_{1} x^{3}}}\, x}{2} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {\frac {c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+4}}{c_{1} x^{3}}}\, x}{2} \\ \end{align*} |
Problem 11123
| ODE |
\[ \boxed {\frac {1}{\sqrt {x^{2}+y^{2}}}+\left (\frac {1}{y}-\frac {x}{y \sqrt {x^{2}+y^{2}}}\right ) y^{\prime }=0} \] |
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program solution |
\[ \ln \left (\sqrt {x^{2}+y^{2}}+x \right ) = c_{1} \] Verified OK.
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Maple solution | \[ -c_{1} +\sqrt {y \left (x \right )^{2}+x^{2}}+x = 0 \] |
Problem 11124
| ODE |
\[ \boxed {y+y^{\prime } x=-x} \] |
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program solution |
\[ y = \frac {-x^{2}+2 c_{1}}{2 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x}{2}+\frac {c_{1}}{x} \] |
Problem 11125
| ODE |
\[ \boxed {-2 y+\left (2 y-2 x -3\right ) y^{\prime }=-6 x -1} \] |
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program solution |
\[ x \left (3 x -2 y+1\right )+y^{2}-3 y = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\sqrt {1-8 \left (x -\frac {1}{2}\right )^{2} c_{1}^{2}}+\left (2 x +3\right ) c_{1}}{2 c_{1}} \] |
Problem 11126
| ODE |
\[ \boxed {\sec \left (x \right ) \cos \left (y\right )^{2}-\cos \left (x \right ) \sin \left (y\right ) y^{\prime }=0} \] |
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program solution |
\[ y = \operatorname {arcsec}\left (\tan \left (x \right )+c_{1} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arccos \left (\frac {1}{\tan \left (x \right )+c_{1}}\right ) \] |
Problem 11127
| ODE |
\[ \boxed {\left (x +1\right ) y^{2}-y^{\prime } x^{3}=0} \] |
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program solution |
\[ y = \frac {2 x^{2}}{c_{3} x^{2}+2 x +1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 x^{2}}{2 c_{1} x^{2}+2 x +1} \] |
Problem 11128
| ODE |
\[ \boxed {2 \left (1-y^{2}\right ) x y+\left (x^{2}+1\right ) \left (1+y^{2}\right ) y^{\prime }=0} \] |
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program solution |
\[ -\frac {\ln \left (x^{2}+1\right )}{2}+\frac {\ln \left (y+1\right )}{2}-\frac {\ln \left (y\right )}{2}+\frac {\ln \left (y-1\right )}{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {c_{1} x^{2}}{2}+\frac {c_{1}}{2}-\frac {\sqrt {4+\left (x^{2}+1\right )^{2} c_{1}^{2}}}{2} \\ y \left (x \right ) &= \frac {c_{1} x^{2}}{2}+\frac {c_{1}}{2}+\frac {\sqrt {4+\left (x^{2}+1\right )^{2} c_{1}^{2}}}{2} \\ \end{align*} |
Problem 11129
| ODE |
\[ \boxed {\sin \left (x \right ) \cos \left (y\right )^{2}+\cos \left (x \right )^{2} y^{\prime }=0} \] |
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program solution |
\[ -\sec \left (x \right )-\tan \left (y\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\arctan \left (\sec \left (x \right )+c_{1} \right ) \] |
Problem 11130
| ODE |
\[ \boxed {x \,{\mathrm e}^{\frac {y}{x}}+y-y^{\prime } x=0} \] |
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program solution |
\[ y = -\ln \left (\ln \left (-\frac {1}{c_{1} x}\right )\right ) x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \ln \left (-\frac {1}{\ln \left (x \right )+c_{1}}\right ) x \] |
Problem 11131
| ODE |
\[ \boxed {2 x^{2} y+3 y^{3}-\left (x^{3}+2 y^{2} x \right ) y^{\prime }=0} \] |
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program solution |
\[ -\ln \left (y\right )-\frac {\ln \left (x^{2}+y^{2}\right )}{2} = -3 \ln \left (x \right )+c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {-2-2 \sqrt {4 c_{1} x^{2}+1}}\, x}{2} \\ y \left (x \right ) &= \frac {\sqrt {-2-2 \sqrt {4 c_{1} x^{2}+1}}\, x}{2} \\ y \left (x \right ) &= -\frac {\sqrt {-2+2 \sqrt {4 c_{1} x^{2}+1}}\, x}{2} \\ y \left (x \right ) &= \frac {\sqrt {-2+2 \sqrt {4 c_{1} x^{2}+1}}\, x}{2} \\ \end{align*} |
Problem 11132
| ODE |
\[ \boxed {y^{2}-x y+y^{\prime } x^{2}=0} \] |
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program solution |
\[ y = \frac {x}{\ln \left (x \right )+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x}{\ln \left (x \right )+c_{1}} \] |
Problem 11133
| ODE |
\[ \boxed {2 x^{2} y+y^{3}-y^{\prime } x^{3}=0} \] |
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program solution |
\[ y = \frac {x^{2}}{\sqrt {-x^{2}+c_{1}}} \] Verified OK. \[ y = -\frac {x^{2}}{\sqrt {-x^{2}+c_{1}}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {x^{2}}{\sqrt {-x^{2}+c_{1}}} \\ y \left (x \right ) &= -\frac {x^{2}}{\sqrt {-x^{2}+c_{1}}} \\ \end{align*} |
Problem 11134
| ODE |
\[ \boxed {y^{3}+y^{\prime } x^{3}=0} \] |
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program solution |
\[ \frac {1}{2 x^{2}}+\frac {1}{2 y^{2}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {x}{\sqrt {c_{1} x^{2}-1}} \\ y \left (x \right ) &= -\frac {x}{\sqrt {c_{1} x^{2}-1}} \\ \end{align*} |
Problem 11135
| ODE |
\[ \boxed {y \cos \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right ) y^{\prime }=-x} \] |
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program solution |
\[ -\ln \left (\frac {1}{x}\right )-\sin \left (\frac {y}{x}\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arcsin \left (\ln \left (x \right )+c_{1} \right ) x \] |
Problem 11136
| ODE |
\[ \boxed {3 y+\left (x +y+1\right ) y^{\prime }=-4 x -1} \] |
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program solution |
\[ y = {\mathrm e}^{\operatorname {LambertW}\left (-\left (x -2\right ) {\mathrm e}^{-c_{1}}\right )+c_{1}}-2 x +1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -3-\frac {\left (x -2\right ) \left (2 \operatorname {LambertW}\left (c_{1} \left (x -2\right )\right )+1\right )}{\operatorname {LambertW}\left (c_{1} \left (x -2\right )\right )} \] |
Problem 11137
| ODE |
\[ \boxed {-y+\left (x +y+3\right ) y^{\prime }=-4 x -2} \] |
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program solution |
\[ \frac {\ln \left (y^{2}+4 x^{2}+4 y+8 x +8\right )}{2}+\frac {\arctan \left (\frac {y+2}{2 x +2}\right )}{2} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -2+\tan \left (\operatorname {RootOf}\left (2 \ln \left (2\right )+\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )-\textit {\_Z} +2 \ln \left (1+x \right )+2 c_{1} \right )\right ) \left (-2 x -2\right ) \] |
Problem 11138
| ODE |
\[ \boxed {y-\left (4 x +2 y-1\right ) y^{\prime }=-2 x} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-\operatorname {LambertW}\left (-2 \,{\mathrm e}^{-25 x +4+25 c_{1}}\right )-25 x +4+25 c_{1}}}{5}-2 x +\frac {2}{5} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\operatorname {LambertW}\left (-2 \,{\mathrm e}^{4-25 x +25 c_{1}}\right )}{10}+\frac {2}{5}-2 x \] |
Problem 11139
| ODE |
\[ \boxed {y+2 y^{2} x -y^{3} x^{2}+2 x^{2} y y^{\prime }=0} \] |
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program solution |
\[ y = \frac {c_{3} -x}{x \left (c_{3} +x \right )} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\tanh \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )}{x} \\ \end{align*} |
Problem 11140
| ODE |
\[ \boxed {2 y+3 y^{2} x +\left (x +2 x^{2} y\right ) y^{\prime }=0} \] |
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program solution |
\[ y x^{2} \left (x y+1\right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {-x +\sqrt {x \left (4 c_{1} +x \right )}}{2 x^{2}} \\ y \left (x \right ) &= \frac {-x -\sqrt {x \left (4 c_{1} +x \right )}}{2 x^{2}} \\ \end{align*} |
Problem 11141
| ODE |
\[ \boxed {y+y^{2} x +\left (x -x^{2} y\right ) y^{\prime }=0} \] |
|
program solution |
\[ y = -\frac {1}{x \operatorname {LambertW}\left (-\frac {{\mathrm e}^{c_{1}}}{x^{2}}\right )} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = -\frac {1}{\operatorname {LambertW}\left (-\frac {c_{1}}{x^{2}}\right ) x} \] |
Problem 11142
| ODE |
\[ \boxed {y^{\prime }+y \cot \left (x \right )=\sec \left (x \right )} \] |
|
program solution |
\[ y = -\frac {\ln \left (\cos \left (x \right )\right )-c_{1}}{\sin \left (x \right )} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \csc \left (x \right ) \left (-\ln \left (\cos \left (x \right )\right )+c_{1} \right ) \] |
Problem 11143
| ODE |
\[ \boxed {y^{\prime } x +y \left (x +1\right )={\mathrm e}^{x}} \] |
|
program solution |
\[ y = \frac {\left ({\mathrm e}^{2 x}+2 c_{1} \right ) {\mathrm e}^{-x}}{2 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{x}+2 c_{1} {\mathrm e}^{-x}}{2 x} \] |
Problem 11144
| ODE |
\[ \boxed {y^{\prime }-\frac {2 y}{x +1}=\left (x +1\right )^{3}} \] |
|
program solution |
\[ y = \frac {\left (x +1\right )^{2} \left (x^{2}+2 c_{1} +2 x \right )}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (x +\frac {1}{2} x^{2}+c_{1} \right ) \left (1+x \right )^{2} \] |
Problem 11145
| ODE |
\[ \boxed {\left (x^{3}+x \right ) y^{\prime }+4 x^{2} y=2} \] |
|
program solution |
\[ y = \frac {x^{2}+2 \ln \left (x \right )+c_{1}}{x^{4}+2 x^{2}+1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {x^{2}+2 \ln \left (x \right )+c_{1}}{\left (x^{2}+1\right )^{2}} \] |
Problem 11146
| ODE |
\[ \boxed {y^{\prime } x^{2}+\left (-2 x +1\right ) y=x^{2}} \] |
|
program solution |
\[ y = x^{2} \left ({\mathrm e}^{-\frac {1}{x}}+c_{1} \right ) {\mathrm e}^{\frac {1}{x}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = x^{2} \left (1+{\mathrm e}^{\frac {1}{x}} c_{1} \right ) \] |
Problem 11147
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }-2 y \left (x +1\right )-y^{\frac {5}{2}}=0} \] |
|
program solution |
\[ y = \frac {2^{\frac {1}{3}} 1024^{\frac {2}{3}} {\left (\left (\left (x -1\right )^{3} \left (\ln \left (\frac {1}{\left (x -1\right )^{\frac {3}{32}}}\right )+\ln \left (\left (x +1\right )^{\frac {3}{32}}\right )\right )-\frac {5}{8}+c_{1} x^{3}+\left (-3 c_{1} -\frac {3}{16}\right ) x^{2}+\left (3 c_{1} +\frac {9}{16}\right ) x -c_{1} \right )^{2}\right )}^{\frac {2}{3}}}{128 \left (-\frac {3 \left (x -1\right )^{3} \ln \left (x -1\right )}{32}+\frac {3 \left (x -1\right )^{3} \ln \left (x +1\right )}{32}+c_{1} x^{3}+\left (-3 c_{1} -\frac {3}{16}\right ) x^{2}+\left (3 c_{1} +\frac {9}{16}\right ) x -c_{1} -\frac {5}{8}\right )^{2}} \] Verified OK. \[ y = -\frac {2^{\frac {1}{3}} 1024^{\frac {2}{3}} {\left (\left (\left (x -1\right )^{3} \left (\ln \left (\frac {1}{\left (x -1\right )^{\frac {3}{32}}}\right )+\ln \left (\left (x +1\right )^{\frac {3}{32}}\right )\right )-\frac {5}{8}+c_{1} x^{3}+\left (-3 c_{1} -\frac {3}{16}\right ) x^{2}+\left (3 c_{1} +\frac {9}{16}\right ) x -c_{1} \right )^{2}\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )^{2}}{512 \left (-\frac {3 \left (x -1\right )^{3} \ln \left (x -1\right )}{32}+\frac {3 \left (x -1\right )^{3} \ln \left (x +1\right )}{32}+c_{1} x^{3}+\left (-3 c_{1} -\frac {3}{16}\right ) x^{2}+\left (3 c_{1} +\frac {9}{16}\right ) x -c_{1} -\frac {5}{8}\right )^{2}} \] Verified OK. \[ y = -\frac {2^{\frac {1}{3}} 1024^{\frac {2}{3}} {\left (\left (\left (x -1\right )^{3} \left (\ln \left (\frac {1}{\left (x -1\right )^{\frac {3}{32}}}\right )+\ln \left (\left (x +1\right )^{\frac {3}{32}}\right )\right )-\frac {5}{8}+c_{1} x^{3}+\left (-3 c_{1} -\frac {3}{16}\right ) x^{2}+\left (3 c_{1} +\frac {9}{16}\right ) x -c_{1} \right )^{2}\right )}^{\frac {2}{3}} \left (i-\sqrt {3}\right )^{2}}{512 \left (-\frac {3 \left (x -1\right )^{3} \ln \left (x -1\right )}{32}+\frac {3 \left (x -1\right )^{3} \ln \left (x +1\right )}{32}+c_{1} x^{3}+\left (-3 c_{1} -\frac {3}{16}\right ) x^{2}+\left (3 c_{1} +\frac {9}{16}\right ) x -c_{1} -\frac {5}{8}\right )^{2}} \] Verified OK.
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Maple solution |
\[ -\frac {-1+\left (-\frac {3 \left (-1+x \right )^{3} \ln \left (-1+x \right )}{32}+\frac {3 \left (-1+x \right )^{3} \ln \left (1+x \right )}{32}+c_{1} x^{3}+\left (-3 c_{1} -\frac {3}{16}\right ) x^{2}+\left (3 c_{1} +\frac {9}{16}\right ) x -c_{1} -\frac {5}{8}\right ) y \left (x \right )^{\frac {3}{2}}}{y \left (x \right )^{\frac {3}{2}}} = 0 \] |
Problem 11148
| ODE |
\[ \boxed {y y^{\prime }+y^{2} x=x} \] |
|
program solution |
\[ -\frac {x^{2}}{2}-\frac {\ln \left (y-1\right )}{2}-\frac {\ln \left (y+1\right )}{2} = c_{1} \] Verified OK.
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|
Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {{\mathrm e}^{-x^{2}} c_{1} +1} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{-x^{2}} c_{1} +1} \\ \end{align*} |
Problem 11149
| ODE |
\[ \boxed {y^{\prime } \sin \left (y\right )+\sin \left (x \right ) \cos \left (y\right )=\sin \left (x \right )} \] |
|
program solution |
\[ \cos \left (x \right )+\ln \left (-1+\cos \left (y\right )\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \arccos \left ({\mathrm e}^{-\cos \left (x \right )} c_{1} +1\right ) \] |
Problem 11150
| ODE |
\[ \boxed {4 y^{\prime } x +3 y+{\mathrm e}^{x} x^{4} y^{5}=0} \] |
|
program solution |
\[ y = \frac {1}{\sqrt {\sqrt {\left ({\mathrm e}^{x}+c_{1} \right ) x}\, x}} \] Verified OK. \[ y = -\frac {1}{\sqrt {\sqrt {\left ({\mathrm e}^{x}+c_{1} \right ) x}\, x}} \] Verified OK. \[ y = -\frac {1}{\sqrt {-\sqrt {\left ({\mathrm e}^{x}+c_{1} \right ) x}\, x}} \] Verified OK. \[ y = \frac {1}{\sqrt {-\sqrt {\left ({\mathrm e}^{x}+c_{1} \right ) x}\, x}} \] Verified OK.
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|
Maple solution |
\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {\sqrt {\left ({\mathrm e}^{x}+c_{1} \right ) x}\, x}} \\ y \left (x \right ) &= \frac {1}{\sqrt {-\sqrt {\left ({\mathrm e}^{x}+c_{1} \right ) x}\, x}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {\sqrt {\left ({\mathrm e}^{x}+c_{1} \right ) x}\, x}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {-\sqrt {\left ({\mathrm e}^{x}+c_{1} \right ) x}\, x}} \\ \end{align*} |
Problem 11151
| ODE |
\[ \boxed {y^{\prime }-\frac {y+1}{x +1}-\sqrt {y+1}=0} \] |
|
program solution |
\[ \ln \left (-x^{2}-2 x +y\right )+\ln \left (\sqrt {y+1}-1-x \right )-\ln \left (\sqrt {y+1}+x +1\right ) = \ln \left (x +1\right )+c_{1} \] Verified OK.
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|
Maple solution | \[ \frac {\left (-c_{1} y \left (x \right )+1+c_{1} x^{2}+\left (2 c_{1} +1\right ) x \right ) \sqrt {y \left (x \right )+1}-\left (1+x \right ) \left (-c_{1} y \left (x \right )-1+c_{1} x^{2}+\left (2 c_{1} -1\right ) x \right )}{\left (x^{2}+2 x -y \left (x \right )\right ) \left (-\sqrt {y \left (x \right )+1}+1+x \right )} = 0 \] |
Problem 11152
| ODE |
\[ \boxed {x^{4} y \left (3 y+2 y^{\prime } x \right )+x^{2} \left (4 y+3 y^{\prime } x \right )=0} \] |
|
program solution |
\[ -\frac {3 \ln \left (y\right )}{2}-\frac {\ln \left (x^{2} y+2\right )}{2} = 2 \ln \left (x \right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}-2 \textit {\_Z}^{2} c_{1} -c_{1} \right )^{6} x^{2}-2 c_{1}}{x^{2} c_{1}} \] |
Problem 11153
| ODE |
\[ \boxed {y^{2} \left (3 y-6 y^{\prime } x \right )-x \left (y-2 y^{\prime } x \right )=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {x}}{3} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {x}}{3} \\ y \left (x \right ) &= c_{1} \sqrt {x} \\ \end{align*} |
Problem 11154
| ODE |
\[ \boxed {2 y x^{3}-y^{2}-\left (2 x^{4}+x y\right ) y^{\prime }=0} \] |
|
program solution |
\[ \frac {\ln \left (y\right )}{2}-\frac {\ln \left (x^{3}+y\right )}{4} = -\frac {\ln \left (x \right )}{4}+c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {c_{1} \left (\sqrt {4 x^{4}+c_{1}^{2}}+c_{1} \right )}{2 x} \\ y \left (x \right ) &= -\frac {c_{1} \left (-c_{1} +\sqrt {4 x^{4}+c_{1}^{2}}\right )}{2 x} \\ \end{align*} |
Problem 11155
| ODE |
\[ \boxed {y^{2}-x y+y^{\prime } x^{2}=0} \] |
|
program solution |
\[ y = \frac {x}{\ln \left (x \right )+c_{3}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {x}{\ln \left (x \right )+c_{1}} \] |
Problem 11156
| ODE |
\[ \boxed {\frac {y^{\prime } x -y}{\sqrt {-y^{2}+x^{2}}}-y^{\prime } x=0} \] |
|
program solution |
\[ \frac {y \left (\ln \left (2\right )+\ln \left (\frac {\sqrt {-y^{2}}\, \sqrt {-y^{2}+x^{2}}-y^{2}}{x}\right )\right )}{\sqrt {-y^{2}}}-y+\frac {\sqrt {-y^{2}}\, \ln \left (y\right )}{y} = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right )-\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )-c_{1} = 0 \] |
Problem 11157
| ODE |
\[ \boxed {y-\left (x -y\right ) y^{\prime }=-x} \] |
|
program solution |
\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}+\arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x \right )+2 c_{1} \right )\right ) x \] |
Problem 11158
| ODE |
\[ \boxed {y^{2}-2 y y^{\prime } x=-x^{2}} \] |
|
program solution |
\[ x -\frac {y^{2}}{x} = c_{1} \] Verified OK.
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|
Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {\left (c_{1} +x \right ) x} \\ y \left (x \right ) &= -\sqrt {\left (c_{1} +x \right ) x} \\ \end{align*} |
Problem 11159
| ODE |
\[ \boxed {-y^{2}+2 y y^{\prime } x=-x} \] |
|
program solution |
\[ \frac {y^{2}}{x}+\ln \left (x \right ) = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {-x \left (\ln \left (x \right )-c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {\left (-\ln \left (x \right )+c_{1} \right ) x} \\ \end{align*} |
Problem 11160
| ODE |
\[ \boxed {y^{\prime } x -y-y^{2}=x^{2}} \] |
|
program solution |
\[ y = \frac {\left (-c_{3} \cos \left (x \right )+\sin \left (x \right )\right ) x}{c_{3} \sin \left (x \right )+\cos \left (x \right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \tan \left (c_{1} +x \right ) x \] |
Problem 11161
| ODE |
\[ \boxed {6 x y+3 y^{2}+\left (2 x^{2}+3 x y\right ) y^{\prime }=-3 x^{2}} \] |
|
program solution |
\[ \frac {3 x^{4}}{4}+2 y x^{3}+\frac {3 x^{2} y^{2}}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {-4 c_{1} x^{2}-\sqrt {-2 c_{1}^{2} x^{4}+6}}{6 c_{1} x} \\ y \left (x \right ) &= \frac {-4 c_{1} x^{2}+\sqrt {-2 c_{1}^{2} x^{4}+6}}{6 c_{1} x} \\ \end{align*} |
Problem 11162
| ODE |
\[ \boxed {\left (x^{2}+y^{2}+2 y\right ) y^{\prime }=-2 x} \] |
|
program solution |
\[ x^{2} {\mathrm e}^{y}+{\mathrm e}^{y} y^{2} = c_{1} \] Verified OK.
|
|
Maple solution | \[ \left (y \left (x \right )^{2}+x^{2}\right ) {\mathrm e}^{y \left (x \right )}+c_{1} = 0 \] |
Problem 11163
| ODE |
\[ \boxed {y^{4}+2 y+\left (y^{3} x +2 y^{4}-4 x \right ) y^{\prime }=0} \] |
|
program solution |
\[ \frac {\left (y^{3}+2\right ) x}{y^{2}}+y^{2} = c_{1} \] Verified OK.
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|
Maple solution | \[ x -\frac {\left (-y \left (x \right )^{2}+c_{1} \right ) y \left (x \right )^{2}}{y \left (x \right )^{3}+2} = 0 \] |
Problem 11164
| ODE |
\[ \boxed {y x^{3}-y^{4}+\left (y^{3} x -x^{4}\right ) y^{\prime }=0} \] |
|
program solution |
\[ y = {\mathrm e}^{c_{1}} x \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) x}{2} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) x}{2} \\ y \left (x \right ) &= x \\ y \left (x \right ) &= c_{1} x \\ \end{align*} |
Problem 11165
| ODE |
\[ \boxed {y^{2}+2 y m x +\left (y^{2} m -m \,x^{2}-2 x y\right ) y^{\prime }=x^{2}} \] |
|
program solution |
\[ -\ln \left (y m -x \right )+\ln \left (x^{2}+y^{2}\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {m -\sqrt {-4 c_{1}^{2} x^{2}-4 c_{1} x +m^{2}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {m +\sqrt {-4 c_{1}^{2} x^{2}-4 c_{1} x +m^{2}}}{2 c_{1}} \\ \end{align*} |
Problem 11166
| ODE |
\[ \boxed {y^{\prime } x -y+2 x^{2} y=x^{3}} \] |
|
program solution |
\[ y = \frac {x \left ({\mathrm e}^{x^{2}}+2 c_{1} \right ) {\mathrm e}^{-x^{2}}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {x}{2}+x \,{\mathrm e}^{-x^{2}} c_{1} \] |
Problem 11167
| ODE |
\[ \boxed {\left (y+x \right ) y^{\prime }=1} \] |
|
program solution |
\[ y = -\operatorname {LambertW}\left (c_{1} {\mathrm e}^{-x -1}\right )-x -1 \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{-x -1}\right )-1-x \] |
Problem 11168
| ODE |
\[ \boxed {y y^{\prime }-y^{\prime } x +y=-x} \] |
|
program solution |
\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}+\arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x \right )+2 c_{1} \right )\right ) x \] |
Problem 11169
| ODE |
\[ \boxed {y^{\prime } x -a y+y^{2} b=c \,x^{2 a}} \] |
|
program solution |
\[ y = \frac {x^{a} \sqrt {-b c}\, \left (c_{3} \cos \left (\frac {x^{a} \sqrt {-b c}}{a}\right )-\sin \left (\frac {x^{a} \sqrt {-b c}}{a}\right )\right )}{b \left (c_{3} \sin \left (\frac {x^{a} \sqrt {-b c}}{a}\right )+\cos \left (\frac {x^{a} \sqrt {-b c}}{a}\right )\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\tanh \left (\frac {x^{a} \sqrt {b}\, \sqrt {c}+i c_{1} a}{a}\right ) \sqrt {c}\, x^{a}}{\sqrt {b}} \] |
Problem 11170
| ODE |
\[ \boxed {x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime }=0} \] |
|
program solution |
\[ \sqrt {-x^{2}+1}-\frac {\left (y-1\right ) \left (y+1\right )}{\sqrt {1-y^{2}}} = c_{1} \] Verified OK.
|
|
Maple solution | \[ \frac {\left (-1+x \right ) \left (1+x \right )}{\sqrt {-x^{2}+1}}+\frac {\left (-1+y \left (x \right )\right ) \left (y \left (x \right )+1\right )}{\sqrt {1-y \left (x \right )^{2}}}+c_{1} = 0 \] |
Problem 11171
| ODE |
\[ \boxed {\sqrt {1-y^{2}}+\sqrt {-x^{2}+1}\, y^{\prime }=0} \] |
|
program solution |
\[ -\arcsin \left (x \right )-\arcsin \left (y\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\sin \left (\arcsin \left (x \right )+c_{1} \right ) \] |
Problem 11172
| ODE |
\[ \boxed {y^{\prime }-x^{2} y=x^{5}} \] |
|
program solution |
\[ y = -\left ({\mathrm e}^{-\frac {x^{3}}{3}} x^{3}+3 \,{\mathrm e}^{-\frac {x^{3}}{3}}-c_{1} \right ) {\mathrm e}^{\frac {x^{3}}{3}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -x^{3}-3+{\mathrm e}^{\frac {x^{3}}{3}} c_{1} \] |
Problem 11173
| ODE |
\[ \boxed {\left (y-x \right )^{2} y^{\prime }=1} \] |
|
program solution |
\[ y-\frac {\ln \left (y-x +1\right )}{2}+\frac {\ln \left (y-1-x \right )}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right )+\frac {\ln \left (y \left (x \right )-x -1\right )}{2}-\frac {\ln \left (y \left (x \right )-x +1\right )}{2}-c_{1} = 0 \] |
Problem 11174
| ODE |
\[ \boxed {y^{\prime } x +y+{\mathrm e}^{x} x^{4} y^{4}=0} \] |
|
program solution |
\[ \frac {3 \,{\mathrm e}^{x} x^{3} y^{3}-1}{3 x^{3} y^{3}} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {1}{\left (3 \,{\mathrm e}^{x}+c_{1} \right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {1+i \sqrt {3}}{2 \left (3 \,{\mathrm e}^{x}+c_{1} \right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {i \sqrt {3}-1}{2 \left (3 \,{\mathrm e}^{x}+c_{1} \right )^{\frac {1}{3}} x} \\ \end{align*} |
Problem 11175
| ODE |
\[ \boxed {\left (1-x \right ) y+\left (1-y\right ) x y^{\prime }=0} \] |
|
program solution |
\[ y = -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{x +c_{1}}}{x}\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\operatorname {LambertW}\left (-\frac {c_{1} {\mathrm e}^{x}}{x}\right ) \] |
Problem 11176
| ODE |
\[ \boxed {\left (y-x \right ) y^{\prime }+y=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_{1}}\right )+c_{1}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {x}{\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_{1}}\right )} \] |
Problem 11177
| ODE |
\[ \boxed {y^{\prime } x -y-\sqrt {x^{2}+y^{2}}=0} \] |
|
program solution |
\[ y = -\frac {{\mathrm e}^{-c_{1}} \left ({\mathrm e}^{2 c_{1}}-x^{2}\right )}{2} \] Verified OK.
|
|
Maple solution | \[ \frac {-c_{1} x^{2}+y \left (x \right )+\sqrt {y \left (x \right )^{2}+x^{2}}}{x^{2}} = 0 \] |
Problem 11178
| ODE |
\[ \boxed {y^{\prime } x -y-\sqrt {-y^{2}+x^{2}}=0} \] |
|
program solution |
\[ y = -\tan \left (-\ln \left (x \right )+c_{1} \right ) \sqrt {\frac {x^{2}}{\tan \left (-\ln \left (x \right )+c_{1} \right )^{2}+1}} \] Verified OK.
|
|
Maple solution | \[ -\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )+\ln \left (x \right )-c_{1} = 0 \] |
Problem 11179
| ODE |
\[ \boxed {x \sin \left (\frac {y}{x}\right )-y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime }=0} \] |
|
program solution |
\[ x \sin \left (\frac {y}{x}\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = x \arcsin \left (\frac {1}{x c_{1}}\right ) \] |
Problem 11180
| ODE |
\[ \boxed {-2 y+\left (2 x -y+4\right ) y^{\prime }=-x -5} \] |
|
program solution |
\[ \frac {3 \ln \left (y-1+x \right )}{2}-\frac {\ln \left (y-3-x \right )}{2} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (i \sqrt {3}-1\right ) \left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (1+x \right )^{2}-1}+27 c_{1} \left (1+x \right )\right )^{\frac {2}{3}}-3 i \sqrt {3}-3+6 \left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (1+x \right )^{2}-1}+27 c_{1} x +27 c_{1} \right )^{\frac {1}{3}} \left (-1+x \right ) c_{1}}{6 \left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (1+x \right )^{2}-1}+27 c_{1} \left (1+x \right )\right )^{\frac {1}{3}} c_{1}} \] |
Problem 11181
| ODE |
\[ \boxed {y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{\frac {3}{2}}}=\frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}} \] |
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program solution |
\[ \int _{}^{x}-\frac {\left (\sqrt {-\textit {\_a}^{2}+1}\, \textit {\_a} +\left (y-1\right ) \textit {\_a}^{2}-y+1\right ) {\mathrm e}^{\frac {\textit {\_a}}{\sqrt {-\textit {\_a}^{2}+1}}}}{\left (-\textit {\_a}^{2}+1\right )^{\frac {5}{2}}}d \textit {\_a} +\left (-\left (\int _{}^{x}\frac {{\mathrm e}^{\frac {\textit {\_a}}{\sqrt {-\textit {\_a}^{2}+1}}}}{\left (-\textit {\_a}^{2}+1\right )^{\frac {3}{2}}}d \textit {\_a} \right )+{\mathrm e}^{\frac {x}{\sqrt {-x^{2}+1}}}\right ) y = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\int \frac {{\mathrm e}^{\frac {x}{\sqrt {-x^{2}+1}}} \left (x +\sqrt {-x^{2}+1}\right )}{\left (-1+x \right )^{2} \left (1+x \right )^{2}}d x +c_{1} \right ) {\mathrm e}^{-\frac {x}{\sqrt {-x^{2}+1}}} \] |
Problem 11182
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime }-x y-y^{2} a x=0} \] |
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program solution |
\[ y = -\frac {1}{a \left (c_{3} \sqrt {x^{2}-1}+1\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{\sqrt {-1+x}\, \sqrt {1+x}\, c_{1} -a} \] |
Problem 11183
| ODE |
\[ \boxed {x y^{2} \left (3 y+y^{\prime } x \right )-2 y+y^{\prime } x=0} \] |
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program solution |
\[ \frac {x^{2} \left (y^{2} x -1\right )}{y} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {c_{1} -\sqrt {4 x^{5}+c_{1}^{2}}}{2 x^{3}} \\ y \left (x \right ) &= \frac {c_{1} +\sqrt {4 x^{5}+c_{1}^{2}}}{2 x^{3}} \\ \end{align*} |
Problem 11184
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+y=\arctan \left (x \right )} \] |
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program solution |
\[ \left (y-\arctan \left (x \right )+1\right ) {\mathrm e}^{\arctan \left (x \right )} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arctan \left (x \right )-1+{\mathrm e}^{-\arctan \left (x \right )} c_{1} \] |
Problem 11185
| ODE |
\[ \boxed {5 x y-3 y^{3}+\left (3 x^{2}-7 y^{2} x \right ) y^{\prime }=0} \] |
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program solution |
\[ 2 x \left (-y^{2}+x \right ) y \sqrt {x y} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (x^{\frac {3}{2}} \textit {\_Z}^{7}-\textit {\_Z}^{3} x^{\frac {5}{2}}-c_{1} \right )^{2} \\ y \left (x \right ) &= \operatorname {RootOf}\left (x^{\frac {3}{2}} \textit {\_Z}^{7}-\textit {\_Z}^{3} x^{\frac {5}{2}}+c_{1} \right )^{2} \\ \end{align*} |
Problem 11186
| ODE |
\[ \boxed {y^{\prime }+y \cos \left (x \right )=\frac {\sin \left (2 x \right )}{2}} \] |
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program solution |
\[ y = {\mathrm e}^{-\sin \left (x \right )} \left (\sin \left (x \right ) {\mathrm e}^{\sin \left (x \right )}-{\mathrm e}^{\sin \left (x \right )}+c_{1} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sin \left (x \right )-1+{\mathrm e}^{-\sin \left (x \right )} c_{1} \] |
Problem 11187
| ODE |
\[ \boxed {y^{2} x +y-y^{\prime } x=0} \] |
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program solution |
\[ y = -\frac {2 x}{x^{2}+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {2 x}{x^{2}-2 c_{1}} \] |
Problem 11188
| ODE |
\[ \boxed {\left (1-x \right ) y-\left (y+1\right ) x y^{\prime }=0} \] |
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program solution |
\[ y = \operatorname {LambertW}\left ({\mathrm e}^{-x -c_{1}} x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \operatorname {LambertW}\left (\frac {{\mathrm e}^{-x} x}{c_{1}}\right ) \] |
Problem 11189
| ODE |
\[ \boxed {3 x^{2} y+\left (x^{3}+y^{2} x^{3}\right ) y^{\prime }=0} \] |
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program solution |
\[ y = \frac {1}{\sqrt {\frac {1}{\operatorname {LambertW}\left (\frac {{\mathrm e}^{-6 c_{1}}}{x^{6}}\right )}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{\sqrt {\frac {1}{\operatorname {LambertW}\left (\frac {c_{1}}{x^{6}}\right )}}} \] |
Problem 11190
| ODE |
\[ \boxed {\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right )-\left (x^{2}+y^{2}+x \right ) \left (y^{\prime } x -y\right )=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -\cot \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +2 \ln \left (2 \csc \left (\textit {\_Z} \right )^{2} x^{2}+\cot \left (\textit {\_Z} \right ) x +x \right )-\ln \left (\csc \left (\textit {\_Z} \right )^{2} x^{2}\right )+2 c_{1} \right )\right ) x \] |
Problem 11191
| ODE |
\[ \boxed {3 y+\left (2 x +3 y-5\right ) y^{\prime }=-2 x +1} \] |
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program solution |
\[ y = -4 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {x}{12}+\frac {c_{1}}{8}-\frac {7}{12}}}{12}\right )-\frac {2 x}{3}-\frac {7}{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {2 x}{3}-4 \operatorname {LambertW}\left (-\frac {c_{1} {\mathrm e}^{\frac {x}{12}-\frac {7}{12}}}{12}\right )-\frac {7}{3} \] |
Problem 11192
| ODE |
\[ \boxed {y^{3}-2 x^{2} y+\left (2 y^{2} x -x^{3}\right ) y^{\prime }=0} \] |
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program solution |
\[ -\frac {y^{2} \left (2 x^{2}-y^{2}\right )^{2}}{8}+\frac {y^{6}}{8} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {\frac {2 c_{1} x^{3}-2 \sqrt {c_{1}^{2} x^{6}+4}}{c_{1} x^{3}}}\, x}{2} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {\frac {c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+4}}{c_{1} x^{3}}}\, x}{2} \\ \end{align*} |
Problem 11193
| ODE |
\[ \boxed {2 y^{2} x^{3}-y+\left (2 y^{3} x^{2}-x \right ) y^{\prime }=0} \] |
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program solution |
\[ \frac {y x^{3}+1}{x y}+y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\left (-{\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+x^{2} 12^{\frac {1}{3}} \left (x^{2}-c_{1} \right )\right ) 12^{\frac {1}{3}}}{6 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} x^{2} \left (x^{2}-c_{1} \right )\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} x^{2} \left (x^{2}-c_{1} \right ) \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right )\right )}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*} |
Problem 11194
| ODE |
\[ \boxed {\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right )+\sqrt {1+x^{2}+y^{2}}\, \left (-y^{\prime } x +y\right )=0} \] |
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program solution |
\[ \arctan \left (\frac {y}{x}\right )-\sqrt {1+x^{2}+y^{2}} = c_{1} \] Verified OK.
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Maple solution | \[ \arctan \left (\frac {x}{y \left (x \right )}\right )+\sqrt {1+x^{2}+y \left (x \right )^{2}}-c_{1} = 0 \] |
Problem 11195
| ODE |
\[ \boxed {{\mathrm e}^{\frac {y}{x}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime }=-1} \] |
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program solution |
\[ \ln \left (x \right )+\int _{}^{\frac {y}{x}}\frac {{\mathrm e}^{\frac {1}{\textit {\_a}}} \left (\textit {\_a} -1\right )}{\left ({\mathrm e}^{\frac {1}{\textit {\_a}}} \left (\textit {\_a} -1\right )+{\mathrm e}^{\textit {\_a}}+1\right ) \textit {\_a}}d \textit {\_a} -c_{1} = 0 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {{\mathrm e}^{\frac {1}{\textit {\_a}}} \left (\textit {\_a} -1\right )}{\textit {\_a} \left (\textit {\_a} \,{\mathrm e}^{\frac {1}{\textit {\_a}}}+{\mathrm e}^{\textit {\_a}}-{\mathrm e}^{\frac {1}{\textit {\_a}}}+1\right )}d \textit {\_a} +\ln \left (x \right )+c_{1} \right ) x \] |
Problem 11196
| ODE |
\[ \boxed {y^{\prime } x +y-\ln \left (x \right ) y^{2}=0} \] |
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program solution |
\[ y = \frac {1}{-c_{3} x +\ln \left (x \right )+1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{1+c_{1} x +\ln \left (x \right )} \] |
Problem 11197
| ODE |
\[ \boxed {x^{3} y^{4}+y^{3} x^{2}+y^{2} x +y+\left (y^{3} x^{4}-y^{2} x^{3}-y x^{3}+x \right ) y^{\prime }=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 11198
| ODE |
\[ \boxed {\left (2 \sqrt {x y}-x \right ) y^{\prime }+y=0} \] |
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program solution |
\[ \frac {\ln \left (y\right ) \sqrt {y}+\sqrt {x}}{\sqrt {y}} = c_{1} \] Verified OK.
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Maple solution | \[ \ln \left (y \left (x \right )\right )+\frac {x}{\sqrt {y \left (x \right ) x}}-c_{1} = 0 \] |
Problem 11199
| ODE |
\[ \boxed {{y^{\prime }}^{2}+\left (y+x \right ) y^{\prime }+x y=0} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-x}}{c_{2}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {x^{2}}{2}+c_{1} \\ y \left (x \right ) &= c_{1} {\mathrm e}^{-x} \\ \end{align*} |
Problem 11200
| ODE |
\[ \boxed {x {y^{\prime }}^{2}-2 y y^{\prime }=x} \] |
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program solution |
\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ y = \frac {c_{1}^{2} x^{2}-1}{2 c_{1}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \frac {-c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align*} |