Problem 8401
| ODE |
\[ \boxed {y^{\prime }-\sqrt {\frac {a y^{2}+y b +c}{a \,x^{2}+b x +c}}=0} \] |
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program solution |
\[ \frac {\sqrt {\frac {a y^{2}+y b +c}{a \,x^{2}+b x +c}}\, \sqrt {a \,x^{2}+b x +c}\, \left (\ln \left (2\right )-\ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{\sqrt {a}}\right )\right )}{\sqrt {a y^{2}+y b +c}\, \sqrt {a}}+\frac {\ln \left (\frac {a y+\frac {b}{2}}{\sqrt {a}}+\sqrt {a y^{2}+y b +c}\right )}{\sqrt {a}} = c_{1} \] Verified OK.
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Maple solution | \[ -\frac {\sqrt {a \,x^{2}+b x +c}\, \left (-\ln \left (2\right )+\ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{\sqrt {a}}\right )\right ) \sqrt {\frac {a y \left (x \right )^{2}+b y \left (x \right )+c}{a \,x^{2}+b x +c}}-\left (c_{1} \sqrt {a}+\ln \left (\frac {2 \sqrt {a y \left (x \right )^{2}+b y \left (x \right )+c}\, \sqrt {a}+2 a y \left (x \right )+b}{\sqrt {a}}\right )-\ln \left (2\right )\right ) \sqrt {a y \left (x \right )^{2}+b y \left (x \right )+c}}{\sqrt {a y \left (x \right )^{2}+b y \left (x \right )+c}\, \sqrt {a}} = 0 \] |
Problem 8402
| ODE |
\[ \boxed {y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}}=0} \] |
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program solution |
\[ \int _{}^{x}-\frac {\sqrt {\frac {y^{3}+1}{\textit {\_a}^{3}+1}}}{\sqrt {y^{3}+1}}d \textit {\_a} +\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {y+1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {y-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {y-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {y+1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {y^{3}+1}} = c_{1} \] Verified OK.
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Maple solution | \[ \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{3}+1}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {\frac {y \left (x \right )^{3}+1}{\textit {\_a}^{3}+1}}d \textit {\_a}}{\sqrt {y \left (x \right )^{3}+1}}+c_{1} = 0 \] |
Problem 8403
| ODE |
\[ \boxed {y^{\prime }-\frac {\sqrt {{| y \left (y-1\right ) \left (-1+a y\right )|}}}{\sqrt {{| x \left (x -1\right ) \left (a x -1\right )|}}}=0} \] |
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program solution |
\[ \int _{}^{x}-\frac {1}{\sqrt {{| \textit {\_a} \left (\textit {\_a} -1\right ) \left (a \textit {\_a} -1\right )|}}}d \textit {\_a} +\int _{0}^{y}\frac {1}{\sqrt {{| \textit {\_a} \left (\textit {\_a} -1\right ) \left (a \textit {\_a} -1\right )|}}}d \textit {\_a} = c_{1} \] Verified OK.
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Maple solution | \[ \int \frac {1}{\sqrt {{| x \left (x -1\right ) \left (a x -1\right )|}}}d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {{| \textit {\_a} \left (\textit {\_a} -1\right ) \left (\textit {\_a} a -1\right )|}}}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 8404
| ODE |
\[ \boxed {y^{\prime }-\frac {\sqrt {1-y^{4}}}{\sqrt {-x^{4}+1}}=0} \] |
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program solution |
\[ \int _{}^{x}-\frac {1}{\sqrt {-\textit {\_a}^{4}+1}}d \textit {\_a} +\frac {\sqrt {-y^{2}+1}\, \sqrt {y^{2}+1}\, \operatorname {EllipticF}\left (y, i\right )}{\sqrt {1-y^{4}}} = c_{1} \] Verified OK.
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Maple solution | \[ \int \frac {1}{\sqrt {-x^{4}+1}}d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-\textit {\_a}^{4}+1}}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 8405
| ODE |
\[ \boxed {y^{\prime }-\sqrt {\frac {a y^{4}+b y^{2}+1}{a \,x^{4}+b \,x^{2}+1}}=0} \] |
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program solution |
\[ \int _{}^{x}-\frac {\sqrt {\frac {a y^{4}+b y^{2}+1}{\textit {\_a}^{4} a +\textit {\_a}^{2} b +1}}}{\sqrt {a y^{4}+b y^{2}+1}}d \textit {\_a} +\frac {2 \sqrt {1-\left (-\frac {b}{2}+\frac {\sqrt {b^{2}-4 a}}{2}\right ) y^{2}}\, \sqrt {1-\left (-\frac {b}{2}-\frac {\sqrt {b^{2}-4 a}}{2}\right ) y^{2}}\, \operatorname {EllipticF}\left (\frac {y \sqrt {-2 b +2 \sqrt {b^{2}-4 a}}}{2}, \sqrt {-1-\frac {b \left (-\frac {b}{2}-\frac {\sqrt {b^{2}-4 a}}{2}\right )}{a}}\right )}{\sqrt {-2 b +2 \sqrt {b^{2}-4 a}}\, \sqrt {a y^{4}+b y^{2}+1}} = c_{1} \] Verified OK.
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Maple solution | \[ \int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \,\textit {\_a}^{4}+\textit {\_a}^{2} b +1}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {\frac {a y \left (x \right )^{4}+b y \left (x \right )^{2}+1}{a \,\textit {\_a}^{4}+\textit {\_a}^{2} b +1}}d \textit {\_a}}{\sqrt {a y \left (x \right )^{4}+b y \left (x \right )^{2}+1}}+c_{1} = 0 \] |
Problem 8406
| ODE |
\[ \boxed {y^{\prime }-\sqrt {\left (b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0} \right ) \left (a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} \right )}=0} \] |
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program solution |
\[ \text {Expression too large to display} \] Warning, solution could not be verified |
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Maple solution | \[ \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{4} b_{4} +\textit {\_a}^{3} b_{3} +\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0}}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {\left (b_{4} y \left (x \right )^{4}+b_{3} y \left (x \right )^{3}+b_{2} y \left (x \right )^{2}+b_{1} y \left (x \right )+b_{0} \right ) \left (\textit {\_a}^{4} a_{4} +\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0} \right )}d \textit {\_a}}{\sqrt {b_{4} y \left (x \right )^{4}+b_{3} y \left (x \right )^{3}+b_{2} y \left (x \right )^{2}+b_{1} y \left (x \right )+b_{0}}}+c_{1} = 0 \] |
Problem 8407
| ODE |
\[ \boxed {y^{\prime }-\sqrt {\frac {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}}=0} \] |
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program solution |
\[ \text {Expression too large to display} \] Warning, solution could not be verified |
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Maple solution | \[ \int _{}^{y \left (x \right )}\sqrt {\textit {\_a}^{4} b_{4} +\textit {\_a}^{3} b_{3} +\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0}}d \textit {\_a} -\sqrt {b_{4} y \left (x \right )^{4}+b_{3} y \left (x \right )^{3}+b_{2} y \left (x \right )^{2}+b_{1} y \left (x \right )+b_{0}}\, \left (\int _{}^{x}\sqrt {\frac {\textit {\_a}^{4} a_{4} +\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}{b_{4} y \left (x \right )^{4}+b_{3} y \left (x \right )^{3}+b_{2} y \left (x \right )^{2}+b_{1} y \left (x \right )+b_{0}}}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 8408
| ODE |
\[ \boxed {y^{\prime }-\sqrt {\frac {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}{a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}}=0} \] |
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program solution |
\[ \text {Expression too large to display} \] Warning, solution could not be verified |
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Maple solution | \[ \int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{4} b_{4} +\textit {\_a}^{3} b_{3} +\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0}}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {\frac {b_{4} y \left (x \right )^{4}+b_{3} y \left (x \right )^{3}+b_{2} y \left (x \right )^{2}+b_{1} y \left (x \right )+b_{0}}{\textit {\_a}^{4} a_{4} +\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}}d \textit {\_a}}{\sqrt {b_{4} y \left (x \right )^{4}+b_{3} y \left (x \right )^{3}+b_{2} y \left (x \right )^{2}+b_{1} y \left (x \right )+b_{0}}}+c_{1} = 0 \] |
Problem 8409
| ODE |
\[ \boxed {y^{\prime }-\operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right ) \operatorname {R2} \left (y, \sqrt {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}\right )=0} \] |
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program solution |
\[ \int _{}^{x}-\operatorname {R1} \left (\textit {\_a} , \sqrt {\textit {\_a}^{4} a_{4} +\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}\right )d \textit {\_a} +\int _{0}^{y}\frac {1}{\operatorname {R2} \left (\textit {\_a} , \sqrt {b_{4} \textit {\_a}^{4}+b_{3} \textit {\_a}^{3}+b_{2} \textit {\_a}^{2}+b_{1} \textit {\_a} +b_{0}}\right )}d \textit {\_a} = c_{1} \] Verified OK.
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Maple solution | \[ \int \operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right )d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\operatorname {R2} \left (\textit {\_a} , \sqrt {\textit {\_a}^{4} b_{4} +\textit {\_a}^{3} b_{3} +\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0}}\right )}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 8410
| ODE |
\[ \boxed {y^{\prime }-\left (\frac {a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{\frac {2}{3}}=0} \] |
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program solution |
\[ \int _{}^{x}-\left (\frac {\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}{a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0}}\right )^{\frac {2}{3}} \left (a_{3} y^{3}+a_{2} y^{2}+a_{1} y+a_{0} \right )^{\frac {2}{3}}d \textit {\_a} +\int _{0}^{y}\left (\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0} \right )^{\frac {2}{3}}d \textit {\_a} = c_{1} \] Verified OK.
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Maple solution | \[ \int _{}^{y \left (x \right )}\left (\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0} \right )^{\frac {2}{3}}d \textit {\_a} -\left (a_{3} y \left (x \right )^{3}+a_{2} y \left (x \right )^{2}+a_{1} y \left (x \right )+a_{0} \right )^{\frac {2}{3}} \left (\int _{}^{x}\left (\frac {\textit {\_a}^{3} a_{3} +\textit {\_a}^{2} a_{2} +\textit {\_a} a_{1} +a_{0}}{a_{3} y \left (x \right )^{3}+a_{2} y \left (x \right )^{2}+a_{1} y \left (x \right )+a_{0}}\right )^{\frac {2}{3}}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 8411
| ODE |
\[ \boxed {y^{\prime }-f \left (x \right ) \left (y-g \left (x \right )\right ) \sqrt {\left (y-a \right ) \left (y-b \right )}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 8412
| ODE |
\[ \boxed {y^{\prime }-{\mathrm e}^{-y+x}=-{\mathrm e}^{x}} \] |
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program solution |
\[ y = \ln \left (-1+{\mathrm e}^{{\mathrm e}^{x}+c_{1}}\right )-{\mathrm e}^{x}-c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -{\mathrm e}^{x}+\ln \left (-1+{\mathrm e}^{{\mathrm e}^{x}+c_{1}}\right )-c_{1} \] |
Problem 8413
| ODE |
\[ \boxed {y^{\prime }-a \cos \left (y\right )=-b} \] |
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program solution |
\[ y = 2 \arctan \left (\frac {\tanh \left (\frac {c_{1} \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}+\frac {x \sqrt {\left (a -b \right ) \left (a +b \right )}}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}{a +b}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 \arctan \left (\frac {\tanh \left (\frac {\sqrt {a^{2}-b^{2}}\, \left (x +c_{1} \right )}{2}\right ) \sqrt {a^{2}-b^{2}}}{a +b}\right ) \] |
Problem 8414
| ODE |
\[ \boxed {y^{\prime }-\cos \left (a y+b x \right )=0} \] |
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program solution |
\[ y = -\frac {2 \arctan \left (\frac {\sin \left (b x \right ) a \cos \left (\frac {\sqrt {-a^{2}+b^{2}}\, \left (a \sqrt {\left (a -b \right ) \left (a +b \right )}\, c_{1} +2 \,\operatorname {arctanh}\left (\frac {\left (a +b \right ) \sin \left (\frac {b x}{2}\right )}{\cos \left (\frac {b x}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) b +2 \sqrt {\left (a -b \right ) \left (a +b \right )}\, \arctan \left (\frac {\sin \left (\frac {b x}{2}\right )}{\cos \left (\frac {b x}{2}\right )}\right )\right )}{2 b \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )+\sin \left (\frac {\sqrt {-a^{2}+b^{2}}\, \left (a \sqrt {\left (a -b \right ) \left (a +b \right )}\, c_{1} +2 \,\operatorname {arctanh}\left (\frac {\left (a +b \right ) \sin \left (\frac {b x}{2}\right )}{\cos \left (\frac {b x}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) b +2 \sqrt {\left (a -b \right ) \left (a +b \right )}\, \arctan \left (\frac {\sin \left (\frac {b x}{2}\right )}{\cos \left (\frac {b x}{2}\right )}\right )\right )}{2 b \sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) \sqrt {-a^{2}+b^{2}}}{\cos \left (\frac {\sqrt {-a^{2}+b^{2}}\, \left (a \sqrt {\left (a -b \right ) \left (a +b \right )}\, c_{1} +2 \,\operatorname {arctanh}\left (\frac {\left (a +b \right ) \sin \left (\frac {b x}{2}\right )}{\cos \left (\frac {b x}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) b +2 \sqrt {\left (a -b \right ) \left (a +b \right )}\, \arctan \left (\frac {\sin \left (\frac {b x}{2}\right )}{\cos \left (\frac {b x}{2}\right )}\right )\right )}{2 b \sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) \left (\cos \left (b x \right ) a -b \right )}\right )}{a} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-b x -2 \arctan \left (\frac {\tanh \left (\frac {\sqrt {a^{2}-b^{2}}\, \left (-x +c_{1} \right )}{2}\right ) \sqrt {a^{2}-b^{2}}}{a -b}\right )}{a} \] |
Problem 8415
| ODE |
\[ \boxed {y^{\prime }+a \sin \left (\alpha y+\beta x \right )=-b} \] |
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program solution |
\[ \text {Expression too large to display} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\beta x +2 \arctan \left (\frac {\tan \left (\frac {\sqrt {\left (-a^{2}+b^{2}\right ) \alpha ^{2}-2 \alpha b \beta +\beta ^{2}}\, \left (-x +c_{1} \right )}{2}\right ) \sqrt {\left (-a^{2}+b^{2}\right ) \alpha ^{2}-2 \alpha b \beta +\beta ^{2}}-a \alpha }{b \alpha -\beta }\right )}{\alpha } \] |
Problem 8416
| ODE |
\[ \boxed {y^{\prime }+f \left (x \right ) \cos \left (a y\right )+g \left (x \right ) \sin \left (a y\right )=-h \left (x \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 8417
| ODE |
\[ \boxed {y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )=f^{\prime }\left (x \right )+1} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -2 \arctan \left (\frac {{\mathrm e}^{\int f \left (x \right )d x}-\left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right ) f \left (x \right )-c_{1} f \left (x \right )}{c_{1} +\int {\mathrm e}^{\int f \left (x \right )d x}d x}\right ) \] |
Problem 8418
| ODE |
\[ \boxed {y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )=1} \] |
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program solution |
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Maple solution | \[ c_{1} +\frac {\tan \left (x \right )}{{\left (\frac {\left (1+\tan \left (y \left (x \right )\right )^{2}\right ) \left (1+\tan \left (x \right )^{2}\right )}{\left (\tan \left (y \left (x \right )\right ) \tan \left (x \right )-1\right )^{2}}\right )}^{\frac {1}{4}}}+\frac {\left (\tan \left (y \left (x \right )\right )+\tan \left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (\tan \left (y \left (x \right )\right )+\tan \left (x \right )\right )^{2}}{\left (\tan \left (y \left (x \right )\right ) \tan \left (x \right )-1\right )^{2}}\right )}{2 \tan \left (y \left (x \right )\right ) \tan \left (x \right )-2} = 0 \] |
Problem 8419
| ODE |
\[ \boxed {y^{\prime }-a \left (1+\tan \left (y\right )^{2}\right )+\tan \left (y\right ) \tan \left (x \right )=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 8420
| ODE |
\[ \boxed {y^{\prime }-\tan \left (y x \right )=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = -i \operatorname {RootOf}\left (-\operatorname {erf}\left (\frac {\left (-x +\textit {\_Z} \right ) \sqrt {2}}{2}\right ) \sqrt {\pi }-\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x +\textit {\_Z} \right )}{2}\right )+c_{1} \sqrt {2}\right ) \] |
Problem 8421
| ODE |
\[ \boxed {y^{\prime }-f \left (a x +y b \right )=0} \] |
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program solution |
\[ -\frac {a x}{b} = \int _{}^{\frac {a x +y b}{b}}-\frac {a}{b f \left (\textit {\_a} b \right )+a}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_a} b \right ) b +a}d \textit {\_a} \right ) b -x +c_{1} \right ) b -a x}{b} \] |
Problem 8422
| ODE |
\[ \boxed {y^{\prime }-x^{-1+a} y^{1-b} f \left (\frac {x^{a}}{a}+\frac {y^{b}}{b}\right )=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = {\left (\frac {\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{-f \left (\frac {\left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{b} a}{a b}\right ) \left (a^{\frac {1}{a}}\right )^{a} \left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{-b} b +f \left (\frac {\left (a^{\frac {1}{a}}\right )^{a} b +\left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{b} a}{a b}\right ) \left (a^{\frac {1}{a}}\right )^{a} \left (\left (\textit {\_a} -b \right )^{\frac {1}{b}}\right )^{-b} \textit {\_a} +a}d \textit {\_a} \right ) a^{2}+a b c_{1} -x^{a} b \right ) a -x^{a} b}{a}\right )}^{\frac {1}{b}} \] |
Problem 8423
| ODE |
\[ \boxed {y^{\prime }-\frac {y-x f \left (x^{2}+a y^{2}\right )}{x +a y f \left (x^{2}+a y^{2}\right )}=0} \] |
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program solution |
\[ -\frac {\arctan \left (\frac {x}{\sqrt {a}\, y}\right )}{\sqrt {a}} = \int _{}^{\frac {x^{2}+a y^{2}}{a}}-\frac {f \left (\textit {\_a} a \right )}{2 \textit {\_a}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \[ \frac {\arctan \left (\frac {\sqrt {a}\, x}{\sqrt {a^{2} y \left (x \right )^{2}}}\right )}{\sqrt {a}}-\frac {\left (\int _{}^{y \left (x \right )^{2}+\frac {x^{2}}{a}}\frac {f \left (\textit {\_a} a \right )}{\textit {\_a}}d \textit {\_a} \right )}{2}-c_{1} = 0 \] |
Problem 8424
| ODE |
\[ \boxed {y^{\prime }-\frac {y a f \left (x^{c} y\right )+c \,x^{a} y^{b}}{x b f \left (x^{c} y\right )-x^{a} y^{b}}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 8425
| ODE |
\[ \boxed {2 y^{\prime }-3 y^{2}-4 a y=b +c \,{\mathrm e}^{-2 a x}} \] |
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program solution |
\[ y = \frac {-\sqrt {3}\, {\mathrm e}^{-a x} \left (\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}-2 a}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )+\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}-2 a}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{3} \right ) \sqrt {c}-\left (\sqrt {4 a^{2}-3 b}+2 a \right ) \left (\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{3} +\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )\right )}{3 \operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{3} +3 \operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\sqrt {3}\, \left (\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}-2 a}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}-2 a}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )\right ) {\mathrm e}^{-a x} \sqrt {c}-\left (\sqrt {4 a^{2}-3 b}+2 a \right ) \left (\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )\right )}{3 \operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +3 \operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )} \] |
Problem 8426
| ODE |
\[ \boxed {x y^{\prime }=\sqrt {a^{2}-x^{2}}} \] |
|
program solution |
\[ y = \sqrt {a^{2}-x^{2}}+\frac {a \ln \left (\sqrt {a^{2}-x^{2}}-a \right )}{2}-\frac {a \ln \left (\sqrt {a^{2}-x^{2}}+a \right )}{2}+c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-x^{2}}+a \right )}{x}\right )+\sqrt {a^{2}-x^{2}}+c_{1} \] |
Problem 8427
| ODE |
\[ \boxed {x y^{\prime }+y=x \sin \left (x \right )} \] |
|
program solution |
\[ y = \frac {-x \cos \left (x \right )+\sin \left (x \right )+c_{1}}{x} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \frac {-\cos \left (x \right ) x +\sin \left (x \right )+c_{1}}{x} \] |
Problem 8428
| ODE |
\[ \boxed {x y^{\prime }-y=\frac {x}{\ln \left (x \right )}} \] |
|
program solution |
\[ y = x \left (\ln \left (\ln \left (x \right )\right )+c_{1} \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \left (\ln \left (\ln \left (x \right )\right )+c_{1} \right ) x \] |
Problem 8429
| ODE |
\[ \boxed {x y^{\prime }-y=x^{2} \sin \left (x \right )} \] |
|
program solution |
\[ y = -x \left (\cos \left (x \right )-c_{1} \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \left (-\cos \left (x \right )+c_{1} \right ) x \] |
Problem 8430
| ODE |
\[ \boxed {x y^{\prime }-y=\frac {x \cos \left (\ln \left (\ln \left (x \right )\right )\right )}{\ln \left (x \right )}} \] |
|
program solution |
\[ y = x \left (\sin \left (\ln \left (\ln \left (x \right )\right )\right )+c_{1} \right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \left (\sin \left (\ln \left (\ln \left (x \right )\right )\right )+c_{1} \right ) x \] |
Problem 8431
| ODE |
\[ \boxed {x y^{\prime }+a y=-b \,x^{n}} \] |
|
program solution |
\[ y = -\frac {\left (b \,x^{n} x^{a}-a c_{1} -c_{1} n \right ) x^{-a}}{a +n} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = -\frac {b \,x^{n}}{a +n}+x^{-a} c_{1} \] |
Problem 8432
| ODE |
\[ \boxed {x y^{\prime }+y^{2}=-x^{2}} \] |
|
program solution |
\[ y = -\frac {\left (c_{3} \operatorname {BesselJ}\left (1, x\right )+\operatorname {BesselY}\left (1, x\right )\right ) x}{c_{3} \operatorname {BesselJ}\left (0, x\right )+\operatorname {BesselY}\left (0, x\right )} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = -\frac {\left (c_{1} \operatorname {BesselY}\left (1, x\right )+\operatorname {BesselJ}\left (1, x\right )\right ) x}{c_{1} \operatorname {BesselY}\left (0, x\right )+\operatorname {BesselJ}\left (0, x\right )} \] |
Problem 8433
| ODE |
\[ \boxed {x y^{\prime }-y^{2}=-1} \] |
|
program solution |
\[ y = \frac {-x^{2}+c_{3}}{x^{2}+c_{3}} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = -\tanh \left (\ln \left (x \right )+c_{1} \right ) \] |
Problem 8434
| ODE |
\[ \boxed {x y^{\prime }+a y^{2}-y=-b \,x^{2}} \] |
|
program solution |
\[ y = \frac {\sqrt {b}\, \left (c_{3} \cos \left (\sqrt {b}\, \sqrt {a}\, x \right )-\sin \left (\sqrt {b}\, \sqrt {a}\, x \right )\right ) x}{\sqrt {a}\, \left (c_{3} \sin \left (\sqrt {b}\, \sqrt {a}\, x \right )+\cos \left (\sqrt {b}\, \sqrt {a}\, x \right )\right )} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = -\frac {\tan \left (\sqrt {a b}\, \left (x +c_{1} \right )\right ) x \sqrt {a b}}{a} \] |
Problem 8435
| ODE |
\[ \boxed {x y^{\prime }+a y^{2}-y b=-c \,x^{2 b}} \] |
|
program solution |
\[ y = \frac {x^{b} \sqrt {a c}\, \left (c_{3} \cos \left (\frac {x^{b} \sqrt {a c}}{b}\right )-\sin \left (\frac {x^{b} \sqrt {a c}}{b}\right )\right )}{a \left (c_{3} \sin \left (\frac {x^{b} \sqrt {a c}}{b}\right )+\cos \left (\frac {x^{b} \sqrt {a c}}{b}\right )\right )} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = -\frac {\tan \left (\frac {x^{b} \sqrt {a}\, \sqrt {c}+c_{1} b}{b}\right ) \sqrt {c}\, x^{b}}{\sqrt {a}} \] |
Problem 8436
| ODE |
\[ \boxed {x y^{\prime }+a y^{2}-y b=c \,x^{\beta }} \] |
|
program solution |
\[ y = \frac {-\sqrt {-a c}\, \left (\operatorname {BesselJ}\left (\frac {b}{\beta }+1, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{3} +\operatorname {BesselY}\left (\frac {b}{\beta }+1, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right ) x^{\frac {\beta }{2}}+b \left (\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )+\operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{3} \right )}{a \left (\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )+\operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\sqrt {-a c}\, \left (\operatorname {BesselY}\left (\frac {b}{\beta }+1, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{\beta }+1, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right ) x^{\frac {\beta }{2}}+b \left (\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\beta }{2}}}{\beta }\right )\right )} \] |
Problem 8437
| ODE |
\[ \boxed {x y^{\prime }+x y^{2}=-a} \] |
|
program solution |
\[ y = \frac {\sqrt {a}\, \left (\operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )+\operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) c_{3} \right )}{\left (\operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right ) c_{3} +\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )\right ) \sqrt {x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sqrt {a}\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) c_{1} +\operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )}{\sqrt {x}\, \left (c_{1} \operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )\right )} \] |
Problem 8438
| ODE |
\[ \boxed {x y^{\prime }+x y^{2}-y=0} \] |
|
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 8439
| ODE |
\[ \boxed {x y^{\prime }+x y^{2}-y=a \,x^{3}} \] |
|
program solution |
\[ y = \frac {x \sqrt {a}\, \left (c_{3} \cosh \left (\frac {x^{2} \sqrt {a}}{2}\right )+\sinh \left (\frac {x^{2} \sqrt {a}}{2}\right )\right )}{c_{3} \sinh \left (\frac {x^{2} \sqrt {a}}{2}\right )+\cosh \left (\frac {x^{2} \sqrt {a}}{2}\right )} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \tanh \left (\sqrt {a}\, \left (\frac {x^{2}}{2}+c_{1} \right )\right ) x \sqrt {a} \] |
Problem 8440
| ODE |
\[ \boxed {x y^{\prime }+x y^{2}-\left (2 x^{2}+1\right ) y=x^{3}} \] |
|
program solution |
\[ y = \frac {x \left (c_{3} \left (1+\sqrt {2}\right ) {\mathrm e}^{\frac {x^{2} \left (1+\sqrt {2}\right )}{2}}-{\mathrm e}^{-\frac {x^{2} \left (\sqrt {2}-1\right )}{2}} \left (\sqrt {2}-1\right )\right )}{c_{3} {\mathrm e}^{\frac {x^{2} \left (1+\sqrt {2}\right )}{2}}+{\mathrm e}^{-\frac {x^{2} \left (\sqrt {2}-1\right )}{2}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x \left (\sqrt {2}+2 \tanh \left (\frac {\left (x^{2}+2 c_{1} \right ) \sqrt {2}}{2}\right )\right ) \sqrt {2}}{2} \] |
Problem 8441
| ODE |
\[ \boxed {x y^{\prime }+a y^{2} x +2 y=-b x} \] |
|
program solution |
\[ y = \frac {\left (x c_{3} \sqrt {-b a}-1\right ) \cosh \left (\sqrt {-b a}\, x \right )+\sinh \left (\sqrt {-b a}\, x \right ) \left (\sqrt {-b a}\, x -c_{3} \right )}{x a \left (c_{3} \sinh \left (\sqrt {-b a}\, x \right )+\cosh \left (\sqrt {-b a}\, x \right )\right )} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \frac {-2 b a c_{1} x -i \sqrt {b}\, \sqrt {a}\, {\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x} x -2 i c_{1} \sqrt {a}\, \sqrt {b}-{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}}{x a \left (2 i c_{1} \sqrt {a}\, \sqrt {b}+{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}\right )} \] |
Problem 8442
| ODE |
\[ \boxed {x y^{\prime }+a y^{2} x +y b=-c x -d} \] |
|
program solution |
\[ y = \frac {\frac {\left (-\left (\frac {1}{2} b^{2}-b \right ) c^{\frac {3}{2}}+i c d \sqrt {a}-\frac {a \,d^{2} \sqrt {c}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}+\frac {c_{3} \left (i c d \sqrt {a}+c^{\frac {3}{2}} b \right ) \operatorname {KummerM}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}-\left (\frac {c^{\frac {3}{2}} b}{2}+i c \sqrt {a}\, \left (c x +\frac {d}{2}\right )\right ) \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )+\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{3} \right )}{c^{\frac {3}{2}} x a \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )+\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{3} \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {4 c \left (a \,c^{3} \left (a d -b \sqrt {-a c}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +c \left (b +2\right )}{2 c}, b +1, 2 x \sqrt {-a c}\right )-\frac {c_{1} \left (a \,d^{2}+b^{2} c \right ) \operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +c \left (b +2\right )}{2 c}, b +1, 2 x \sqrt {-a c}\right )}{4}+a \,c^{3} \left (b \sqrt {-a c}+a d \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +b c}{2 c}, b +1, 2 x \sqrt {-a c}\right )-\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +b c}{2 c}, b +1, 2 x \sqrt {-a c}\right ) c_{1} \left (b c -\sqrt {-a c}\, d \right )}{2}\right )}{4 a^{2} c^{3} \left (\sqrt {-a c}\, d +b c \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +c \left (b +2\right )}{2 c}, b +1, 2 x \sqrt {-a c}\right )-c_{1} \sqrt {-a c}\, \left (a \,d^{2}+b^{2} c \right ) \operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +c \left (b +2\right )}{2 c}, b +1, 2 x \sqrt {-a c}\right )+4 c \left (a^{2} c^{2} \left (b c -\sqrt {-a c}\, d \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +b c}{2 c}, b +1, 2 x \sqrt {-a c}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +b c}{2 c}, b +1, 2 x \sqrt {-a c}\right ) c_{1} \left (b \sqrt {-a c}+a d \right )}{2}\right )} \] |
Problem 8443
| ODE |
\[ \boxed {x y^{\prime }+x^{a} y^{2}+\frac {\left (a -b \right ) y}{2}=-x^{b}} \] |
|
program solution |
\[ y = \frac {\sqrt {x^{a +b}}\, \left (c_{3} \cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )-\sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )\right ) x^{-a}}{c_{3} \sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )+\cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\tan \left (\frac {2 x^{\frac {a}{2}+\frac {b}{2}}+c_{1} \left (a +b \right )}{a +b}\right ) x^{-\frac {a}{2}+\frac {b}{2}} \] |
Problem 8444
| ODE |
\[ \boxed {x y^{\prime }+a \,x^{\alpha } y^{2}+y b=c \,x^{\beta }} \] |
|
program solution |
\[ y = -\frac {c \,x^{\beta } \left (\operatorname {BesselJ}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{3} +\operatorname {BesselY}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right )}{x^{\frac {\alpha }{2}+\frac {\beta }{2}} \left (\operatorname {BesselJ}\left (\frac {b +2 \beta +\alpha }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{3} +\operatorname {BesselY}\left (\frac {b +2 \beta +\alpha }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right ) \sqrt {-a c}-\left (b +\beta \right ) \left (\operatorname {BesselJ}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{3} +\operatorname {BesselY}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c \,x^{\beta } \left (\operatorname {BesselY}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right )}{-\left (\operatorname {BesselY}\left (\frac {b +2 \beta +\alpha }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +2 \beta +\alpha }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right ) x^{\frac {\alpha }{2}+\frac {\beta }{2}} \sqrt {-a c}+\left (b +\beta \right ) \left (\operatorname {BesselY}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right )} \] |
Problem 8445
| ODE |
\[ \boxed {x y^{\prime }-y^{2} \ln \left (x \right )+y=0} \] |
|
program solution |
\[ y = \frac {1}{-x c_{3} +\ln \left (x \right )+1} \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = \frac {1}{1+c_{1} x +\ln \left (x \right )} \] |
Problem 8446
| ODE |
\[ \boxed {x y^{\prime }-y \left (2 y \ln \left (x \right )-1\right )=0} \] |
|
program solution |
\[ y = \frac {1}{-2 x c_{3} +2 \ln \left (x \right )+2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {1}{2+c_{1} x +2 \ln \left (x \right )} \] |
Problem 8447
| ODE |
\[ \boxed {x y^{\prime }+f \left (x \right ) \left (y^{2}-x^{2}\right )=0} \] |
|
program solution |
\[ y = \frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right ) x +\left (-f^{\prime }\left (x \right ) x +f \left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y} \left (x \right ) x}{f \left (x \right ) x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) x}{f \left (x \right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) f \left (x \right ) x +\left (-f^{\prime }\left (x \right ) x +f \left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )-f \left (x \right )^{3} \textit {\_Y} \left (x \right ) x}{f \left (x \right ) x}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 8448
| ODE |
\[ \boxed {x y^{\prime }+y^{3}+3 x y^{2}=0} \] |
|
program solution |
|
|
Maple solution | \[ \frac {3 \,\operatorname {erf}\left (\frac {i \left (3 x y \left (x \right )-1\right ) \sqrt {2}}{2 y \left (x \right )}\right ) \sqrt {2}\, \sqrt {\pi }\, x -2 i {\mathrm e}^{\frac {\left (3 x y \left (x \right )-1\right )^{2}}{2 y \left (x \right )^{2}}}+6 c_{1} x}{6 x} = 0 \] |
Problem 8449
| ODE |
\[ \boxed {x y^{\prime }-\sqrt {y^{2}+x^{2}}-y=0} \] |
|
program solution |
\[ y = -\frac {{\mathrm e}^{-c_{1}} \left ({\mathrm e}^{2 c_{1}}-x^{2}\right )}{2} \] Verified OK.
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|
Maple solution | \[ \frac {-c_{1} x^{2}+y \left (x \right )+\sqrt {y \left (x \right )^{2}+x^{2}}}{x^{2}} = 0 \] |
Problem 8450
| ODE |
\[ \boxed {x y^{\prime }+a \sqrt {y^{2}+x^{2}}-y=0} \] |
|
program solution |
\[ y = \frac {x \left ({\mathrm e}^{-2 a \left (\ln \left (x \right )-c_{1} \right )}-1\right ) {\mathrm e}^{a \left (\ln \left (x \right )-c_{1} \right )}}{2} \] Verified OK.
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|
Maple solution | \[ \frac {x^{a} y \left (x \right )+x^{a} \sqrt {y \left (x \right )^{2}+x^{2}}-c_{1} x}{x} = 0 \] |
Problem 8451
| ODE |
\[ \boxed {x y^{\prime }-\sqrt {y^{2}+x^{2}}\, x -y=0} \] |
|
program solution |
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|
Maple solution | \[ \ln \left (\sqrt {y \left (x \right )^{2}+x^{2}}+y \left (x \right )\right )-x -\ln \left (x \right )-c_{1} = 0 \] |
Problem 8452
| ODE |
\[ \boxed {x y^{\prime }-x \left (y-x \right ) \sqrt {y^{2}+x^{2}}-y=0} \] |
|
program solution |
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|
Maple solution | \[ \ln \left (2\right )+\ln \left (\frac {x \left (\sqrt {2 y \left (x \right )^{2}+2 x^{2}}+y \left (x \right )+x \right )}{y \left (x \right )-x}\right )+\frac {\sqrt {2}\, x^{2}}{2}-\ln \left (x \right )-c_{1} = 0 \] |
Problem 8453
| ODE |
\[ \boxed {x y^{\prime }-x \,{\mathrm e}^{\frac {y}{x}}-y=x} \] |
|
program solution |
\[ y = \ln \left (x \right ) x +x \ln \left (\frac {1}{{\mathrm e}^{c_{1}}-x}\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (\ln \left (-\frac {x}{-1+x \,{\mathrm e}^{c_{1}}}\right )+c_{1} \right ) x \] |
Problem 8454
| ODE |
\[ \boxed {x y^{\prime }-y \ln \left (y\right )=0} \] |
|
program solution |
\[ y = {\mathrm e}^{{\mathrm e}^{c_{1}} x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = {\mathrm e}^{c_{1} x} \] |
Problem 8455
| ODE |
\[ \boxed {x y^{\prime }-y \left (\ln \left (y x \right )-1\right )=0} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{{\mathrm e}^{c_{1}} x}}{x} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{\frac {x}{c_{1}}}}{x} \] |
Problem 8456
| ODE |
\[ \boxed {x y^{\prime }-y \left (x \ln \left (\frac {x^{2}}{y}\right )+2\right )=0} \] |
|
program solution |
|
|
Maple solution | \[ y \left (x \right ) = x^{2} {\mathrm e}^{-{\mathrm e}^{-x} c_{1}} \] |
Problem 8457
| ODE |
\[ \boxed {x y^{\prime }-\sin \left (-y+x \right )=0} \] |
|
program solution |
|
|
Maple solution | \[ \text {No solution found} \] |
Problem 8458
| ODE |
\[ \boxed {x y^{\prime }+\left (\sin \left (y\right )-3 x^{2} \cos \left (y\right )\right ) \cos \left (y\right )=0} \] |
|
program solution |
\[ x \left (-x^{2}+\tan \left (y\right )\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \arctan \left (\frac {x^{3}+2 c_{1}}{x}\right ) \] |
Problem 8459
| ODE |
\[ \boxed {x y^{\prime }-x \sin \left (\frac {y}{x}\right )-y=0} \] |
|
program solution |
\[ y = \arctan \left (-\frac {2 x c_{1}}{c_{1}^{2} x^{2}+1}, -\frac {c_{1}^{2} x^{2}-1}{c_{1}^{2} x^{2}+1}\right ) x \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \arctan \left (\frac {2 x c_{1}}{c_{1}^{2} x^{2}+1}, \frac {-c_{1}^{2} x^{2}+1}{c_{1}^{2} x^{2}+1}\right ) x \] |
Problem 8460
| ODE |
\[ \boxed {x y^{\prime }+x \cos \left (\frac {y}{x}\right )-y=-x} \] |
|
program solution |
\[ y = 2 \arctan \left (-\ln \left (x \right )+c_{1} \right ) x \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -2 \arctan \left (\ln \left (x \right )+c_{1} \right ) x \] |
Problem 8461
| ODE |
\[ \boxed {x y^{\prime }+x \tan \left (\frac {y}{x}\right )-y=0} \] |
|
program solution |
\[ y = -\arcsin \left (\frac {1}{x c_{1}}\right ) x \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = x \arcsin \left (\frac {1}{c_{1} x}\right ) \] |
Problem 8462
| ODE |
\[ \boxed {x y^{\prime }-y f \left (y x \right )=0} \] |
|
program solution |
\[ -\ln \left (x \right ) = \int _{}^{y x}-\frac {1}{\textit {\_a} \left (f \left (\textit {\_a} \right )+1\right )}d \textit {\_a} +c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} +\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \left (1+f \left (\textit {\_a} \right )\right )}d \textit {\_a} \right )}{x} \] |
Problem 8463
| ODE |
\[ \boxed {x y^{\prime }-y f \left (x^{a} y^{b}\right )=0} \] |
|
program solution |
\[ -\frac {a \ln \left (x \right )}{b} = \int _{}^{y x^{\frac {a}{b}}}-\frac {a}{\textit {\_a} \left (b f \left (\textit {\_a}^{b}\right )+a \right )}d \textit {\_a} +c_{1} \] Verified OK.
|
|
Maple solution | \[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {1}{\left (f \left (x^{a} \textit {\_a}^{b}\right ) b +a \right ) \textit {\_a}}d \textit {\_a} -\frac {\ln \left (x \right )}{b}-c_{1} = 0 \] |
Problem 8464
| ODE |
\[ \boxed {x y^{\prime }+a y-f \left (x \right ) g \left (x^{a} y\right )=0} \] |
|
program solution |
|
|
Maple solution | \[ y \left (x \right ) = \operatorname {RootOf}\left (-\left (\int f \left (x \right ) x^{a -1}d x \right )+\int _{}^{\textit {\_Z}}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) x^{-a} \] |
Problem 8465
| ODE |
\[ \boxed {\left (x +1\right ) y^{\prime }+y \left (y-x \right )=0} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{x}}{-{\mathrm e}^{-1} \left (x +1\right ) \operatorname {expIntegral}_{1}\left (-x -1\right )-{\mathrm e}^{x}+\left (x +1\right ) c_{3}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{x}}{-{\mathrm e}^{-1} \left (x +1\right ) \operatorname {expIntegral}_{1}\left (-x -1\right )-{\mathrm e}^{x}+c_{1} \left (x +1\right )} \] |
Problem 8466
| ODE |
\[ \boxed {2 x y^{\prime }-y=2 x^{3}} \] |
|
program solution |
\[ y = \frac {2 x^{3}}{5}+\frac {c_{1} \sqrt {x}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {2 x^{3}}{5}+c_{1} \sqrt {x} \] |
Problem 8467
| ODE |
\[ \boxed {\left (2 x +1\right ) y^{\prime }-4 \,{\mathrm e}^{-y}=-2} \] |
|
program solution |
\[ y = -\ln \left (\frac {2 x +1}{-1+4 \,{\mathrm e}^{2 c_{1}} x +2 \,{\mathrm e}^{2 c_{1}}}\right )-2 c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\ln \left (\frac {2 x +1}{-1+\left (4 x +2\right ) {\mathrm e}^{2 c_{1}}}\right )-2 c_{1} \] |
Problem 8468
| ODE |
\[ \boxed {3 x y^{\prime }-3 x \ln \left (x \right ) y^{4}-y=0} \] |
|
program solution |
\[ -\frac {x}{y^{3}}-\frac {3 \ln \left (x \right ) x^{2}}{2}+\frac {3 x^{2}}{4} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_{1} \right )^{2}\right )}^{\frac {1}{3}}}{6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_{1}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_{1} \right )^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{12 x^{2} \ln \left (x \right )-6 x^{2}-8 c_{1}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (-x \left (6 x^{2} \ln \left (x \right )-3 x^{2}-4 c_{1} \right )^{2}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{12 x^{2} \ln \left (x \right )-6 x^{2}-8 c_{1}} \\ \end{align*} |
Problem 8469
| ODE |
\[ \boxed {x^{2} y^{\prime }+y=x} \] |
|
program solution |
\[ y = {\mathrm e}^{\frac {1}{x}} \left (\operatorname {expIntegral}_{1}\left (\frac {1}{x}\right )+c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (\operatorname {expIntegral}_{1}\left (\frac {1}{x}\right )+c_{1} \right ) {\mathrm e}^{\frac {1}{x}} \] |
Problem 8470
| ODE |
\[ \boxed {x^{2} y^{\prime }-y=-x^{2} {\mathrm e}^{x -\frac {1}{x}}} \] |
|
program solution |
\[ y = -\left ({\mathrm e}^{x}-c_{1} \right ) {\mathrm e}^{-\frac {1}{x}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (-{\mathrm e}^{x}+c_{1} \right ) {\mathrm e}^{-\frac {1}{x}} \] |
Problem 8471
| ODE |
\[ \boxed {x^{2} y^{\prime }-\left (x -1\right ) y=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\frac {\ln \left (x \right ) x +x c_{1} +1}{x}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{\frac {1}{x}} \] |
Problem 8472
| ODE |
\[ \boxed {x^{2} y^{\prime }+y^{2}+y x=-x^{2}} \] |
|
program solution |
\[ y = -\frac {\left (-1+\ln \left (x \right )+c_{3} \right ) x}{\ln \left (x \right )+c_{3}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {x \left (\ln \left (x \right )+c_{1} -1\right )}{\ln \left (x \right )+c_{1}} \] |
Problem 8473
| ODE |
\[ \boxed {x^{2} y^{\prime }-y^{2}-y x=0} \] |
|
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 8474
| ODE |
\[ \boxed {x^{2} y^{\prime }-y^{2}-y x=x^{2}} \] |
|
program solution |
\[ y = \frac {x \left (-\cos \left (\ln \left (x \right )\right ) c_{3} +\sin \left (\ln \left (x \right )\right )\right )}{\sin \left (\ln \left (x \right )\right ) c_{3} +\cos \left (\ln \left (x \right )\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \tan \left (\ln \left (x \right )+c_{1} \right ) x \] |
Problem 8475
| ODE |
\[ \boxed {x^{2} \left (y^{\prime }+y^{2}\right )=-a \,x^{k}+b \left (b -1\right )} \] |
|
program solution |
\[ y = \frac {-\left (\operatorname {BesselJ}\left (\frac {2 b +k -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{3} +\operatorname {BesselY}\left (\frac {2 b +k -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right ) x^{\frac {k}{2}} \sqrt {a}+b \left (\operatorname {BesselJ}\left (\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{3} +\operatorname {BesselY}\left (\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}{x \left (\operatorname {BesselJ}\left (\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{3} +\operatorname {BesselY}\left (\frac {2 b -1}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {-x^{\frac {k}{2}} \left (\operatorname {BesselY}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right ) \sqrt {a}+\left (\frac {1}{2}+\operatorname {csgn}\left (2 b -1\right ) b -\frac {\operatorname {csgn}\left (2 b -1\right )}{2}\right ) \left (\operatorname {BesselY}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}{x \left (\operatorname {BesselY}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )} \] |
Problem 8476
| ODE |
\[ \boxed {x^{2} \left (y^{\prime }+y^{2}\right )+4 y x=-2} \] |
|
program solution |
\[ y = \frac {-c_{3} x -2}{x \left (c_{3} x +1\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {-2 c_{1} +x}{x \left (-x +c_{1} \right )} \] |
Problem 8477
| ODE |
\[ \boxed {x^{2} \left (y^{\prime }+y^{2}\right )+y a x=-b} \] |
|
program solution |
\[ y = \frac {\left (1-a -\sqrt {a^{2}-2 a -4 b +1}\right ) x^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}}-x^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} c_{3} \left (-1+a -\sqrt {a^{2}-2 a -4 b +1}\right )}{2 x \left (x^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} c_{3} +x^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}}\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {1-a +\tanh \left (\frac {\sqrt {a^{2}-2 a -4 b +1}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {a^{2}-2 a -4 b +1}}{2 x} \] |
Problem 8478
| ODE |
\[ \boxed {x^{2} \left (y^{\prime }-y^{2}\right )-a \,x^{2} y=-a x -2} \] |
|
program solution |
\[ y = \frac {-c_{3} \left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}+1}{x \left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x} c_{3} +1\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {-\left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}+c_{1}}{\left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x}+c_{1} \right ) x} \] |
Problem 8479
| ODE |
\[ \boxed {x^{2} \left (y^{\prime }+a y^{2}\right )=b} \] |
|
program solution |
\[ y = \frac {\left (1-\sqrt {4 b a +1}\right ) x^{-\frac {\sqrt {4 b a +1}}{2}}+c_{3} x^{\frac {\sqrt {4 b a +1}}{2}} \left (1+\sqrt {4 b a +1}\right )}{2 x a \left (c_{3} x^{\frac {\sqrt {4 b a +1}}{2}}+x^{-\frac {\sqrt {4 b a +1}}{2}}\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {1+\tanh \left (\frac {\sqrt {4 a b +1}\, \left (\ln \left (x \right )-c_{1} \right )}{2}\right ) \sqrt {4 a b +1}}{2 a x} \] |
Problem 8480
| ODE |
\[ \boxed {x^{2} \left (y^{\prime }+a y^{2}\right )=-b \,x^{\alpha }-c} \] |
|
program solution |
\[ y = \frac {-2 \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) \sqrt {b a}\, x^{\frac {\alpha }{2}}+\left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) \left (\sqrt {-4 a c +1}+1\right )}{2 x a \left (\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{3} +\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {b a}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {-2 \sqrt {a b}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }+1, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) x^{\frac {\alpha }{2}}+\left (\sqrt {-4 a c +1}+1\right ) \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )}{2 x a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )} \] |
Problem 8481
| ODE |
\[ \boxed {x^{2} y^{\prime }+a y^{3}-x^{2} a y^{2}=0} \] |
|
program solution |
|
|
Maple solution | \[ y \left (x \right ) = \frac {1}{-a x -2^{\frac {2}{3}} \left (-a \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (\frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right )+\operatorname {AiryBi}\left (1, \frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right ) c_{1} +\operatorname {AiryAi}\left (1, \frac {\left (\textit {\_Z}^{2} 2^{\frac {1}{3}} \left (-a \right )^{\frac {1}{3}} x -1\right ) 2^{\frac {2}{3}}}{2 \left (-a \right )^{\frac {1}{3}} x}\right )\right )} \] |
Problem 8482
| ODE |
\[ \boxed {x^{2} y^{\prime }+y^{3} x +a y^{2}=0} \] |
|
program solution |
|
|
Maple solution | \[ c_{1} +{\mathrm e}^{-\frac {\left (\left (-x +a \right ) y \left (x \right )+x \right ) \left (\left (a +x \right ) y \left (x \right )+x \right )}{2 y \left (x \right )^{2} x^{2}}} x +\frac {\operatorname {erf}\left (\frac {\sqrt {2}\, \left (a y \left (x \right )+x \right )}{2 y \left (x \right ) x}\right ) \sqrt {2}\, \sqrt {\pi }\, a \,{\mathrm e}^{\frac {1}{2}}}{2} = 0 \] |
Problem 8483
| ODE |
\[ \boxed {x^{2} y^{\prime }+x^{2} y^{3} a +b y^{2}=0} \] |
|
program solution |
|
|
Maple solution | \[ y \left (x \right ) = -\frac {2^{\frac {1}{3}} a b x}{2^{\frac {1}{3}} a \,b^{2}-2 \left (b^{2} a^{2}\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\frac {-a 2^{\frac {2}{3}} x +2 \textit {\_Z}^{2} \left (b^{2} a^{2}\right )^{\frac {1}{3}}}{2 \left (b^{2} a^{2}\right )^{\frac {1}{3}}}\right ) c_{1} \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (\frac {-a 2^{\frac {2}{3}} x +2 \textit {\_Z}^{2} \left (b^{2} a^{2}\right )^{\frac {1}{3}}}{2 \left (b^{2} a^{2}\right )^{\frac {1}{3}}}\right )+\operatorname {AiryBi}\left (1, \frac {-a 2^{\frac {2}{3}} x +2 \textit {\_Z}^{2} \left (b^{2} a^{2}\right )^{\frac {1}{3}}}{2 \left (b^{2} a^{2}\right )^{\frac {1}{3}}}\right ) c_{1} +\operatorname {AiryAi}\left (1, \frac {-a 2^{\frac {2}{3}} x +2 \textit {\_Z}^{2} \left (b^{2} a^{2}\right )^{\frac {1}{3}}}{2 \left (b^{2} a^{2}\right )^{\frac {1}{3}}}\right )\right ) x} \] |
Problem 8484
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+y x=1} \] |
|
program solution |
\[ y = \frac {\operatorname {arcsinh}\left (x \right )+c_{1}}{\sqrt {x^{2}+1}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\operatorname {arcsinh}\left (x \right )+c_{1}}{\sqrt {x^{2}+1}} \] |
Problem 8485
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+y x=x \left (x^{2}+1\right )} \] |
|
program solution |
\[ y = \frac {x^{2} \sqrt {x^{2}+1}+\sqrt {x^{2}+1}+3 c_{1}}{3 \sqrt {x^{2}+1}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {x^{2}}{3}+\frac {1}{3}+\frac {c_{1}}{\sqrt {x^{2}+1}} \] |
Problem 8486
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+2 y x=2 x^{2}} \] |
|
program solution |
\[ y = \frac {2 x^{3}+3 c_{1}}{3 x^{2}+3} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {2 x^{3}+3 c_{1}}{3 x^{2}+3} \] |
Problem 8487
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+\left (y^{2}+1\right ) \left (2 y x -1\right )=0} \] |
|
program solution |
|
|
Maple solution | \[ c_{1} +\frac {x}{{\left (\frac {\left (x^{2}+1\right ) \left (y \left (x \right )^{2}+1\right )}{\left (x y \left (x \right )-1\right )^{2}}\right )}^{\frac {1}{4}}}+\frac {\left (x +y \left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x +y \left (x \right )\right )^{2}}{\left (x y \left (x \right )-1\right )^{2}}\right )}{2 x y \left (x \right )-2} = 0 \] |
Problem 8488
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2}=0} \] |
|
program solution |
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Maple solution | \[ y \left (x \right ) = \frac {\arctan \left (\frac {6 \sqrt {x^{2}+1}\, \left (\sqrt {x^{2}+1}\, x^{2}+\sqrt {x^{2}+1}+3 c_{1} \right )}{10+6 c_{1} \left (x^{2}+1\right )^{\frac {3}{2}}+x^{6}+3 x^{4}+12 x^{2}+9 c_{1}^{2}}, \frac {8+6 \left (-x^{2}-1\right ) c_{1} \sqrt {x^{2}+1}-x^{6}-3 x^{4}+6 x^{2}-9 c_{1}^{2}}{10+6 c_{1} \left (x^{2}+1\right )^{\frac {3}{2}}+x^{6}+3 x^{4}+12 x^{2}+9 c_{1}^{2}}\right )}{2} \] |
Problem 8489
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }-y x=-a} \] |
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program solution |
\[ y = c_{1} \sqrt {x^{2}-1}+a x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sqrt {x -1}\, \sqrt {x +1}\, c_{1} +a x \] |
Problem 8490
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }+2 y x=\cos \left (x \right )} \] |
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program solution |
\[ y = \frac {\sin \left (x \right )+c_{1}}{x^{2}-1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sin \left (x \right )+c_{1}}{x^{2}-1} \] |
Problem 8491
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }+y^{2}-2 y x=-1} \] |
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program solution |
\[ y = \frac {\ln \left (x +1\right ) x -\ln \left (x -1\right ) x +x c_{3} -2}{\ln \left (x +1\right )-\ln \left (x -1\right )+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x +\frac {1}{c_{1} -\operatorname {arctanh}\left (x \right )} \] |
Problem 8492
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }-y \left (y-x \right )=0} \] |
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program solution |
\[ y = \frac {1}{c_{3} \sqrt {x^{2}-1}+x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{\sqrt {x -1}\, \sqrt {x +1}\, c_{1} +x} \] |
Problem 8493
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }+a \left (y^{2}-2 y x +1\right )=0} \] |
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program solution |
\[ y = \frac {\operatorname {LegendreP}\left (a , x\right ) c_{3} +\operatorname {LegendreQ}\left (a , x\right )}{\operatorname {LegendreQ}\left (-1+a , x\right )+\operatorname {LegendreP}\left (-1+a , x\right ) c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 a} \left (-\frac {\left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 a} \left (x +1\right ) \left (x -1\right )^{2} \operatorname {HeunCPrime}\left (0, 2 a -1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{x +1}\right )}{8}-\left (x -1\right )^{2} \operatorname {HeunCPrime}\left (0, -2 a +1, 0, 0, a^{2}-a +\frac {1}{2}, \frac {2}{x +1}\right ) c_{1} +\left (c_{1} \left (\frac {x +1}{x -1}\right )^{-a} \left (\left (a -\frac {1}{2}\right ) x -\frac {a}{2}+\frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [-a +1, -a +1\right ], \left [-2 a +2\right ], -\frac {2}{x -1}\right )+\frac {\operatorname {hypergeom}\left (\left [a , a\right ], \left [2 a \right ], -\frac {2}{x -1}\right ) \left (\frac {x +1}{x -1}\right )^{a} a \left (-\frac {x}{2}-\frac {1}{2}\right )^{-2 a} \left (x -1\right )}{16}\right ) \left (x +1\right )^{2}\right ) \left (\frac {x +1}{x -1}\right )^{a}}{a \left (x +1\right )^{2} \left (c_{1} \operatorname {hypergeom}\left (\left [-a +1, -a +1\right ], \left [-2 a +2\right ], -\frac {2}{x -1}\right ) \left (-\frac {x}{2}-\frac {1}{2}\right )^{2 a}+\frac {\operatorname {hypergeom}\left (\left [a , a\right ], \left [2 a \right ], -\frac {2}{x -1}\right ) \left (\frac {x +1}{x -1}\right )^{2 a} \left (x -1\right )}{8}\right )} \] |
Problem 8494
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }+a y^{2} x +y x=0} \] |
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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 | \[ y \left (x \right ) = \frac {1}{\sqrt {x -1}\, \sqrt {x +1}\, c_{1} -a} \] |
Problem 8495
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime }-2 \ln \left (y\right ) y x=0} \] |
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program solution |
\[ y = {\mathrm e}^{{\mathrm e}^{2 c_{1}} \left (x -1\right ) \left (x +1\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{c_{1} \left (x +1\right ) \left (x -1\right )} \] |
Problem 8496
| ODE |
\[ \boxed {\left (x^{2}-4\right ) y^{\prime }+\left (2+x \right ) y^{2}-4 y=0} \] |
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program solution |
\[ y = \frac {x -2}{\left (\ln \left (2+x \right )+c_{3} \right ) \left (2+x \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x -2}{\left (\ln \left (x +2\right )+c_{1} \right ) \left (x +2\right )} \] |
Problem 8497
| ODE |
\[ \boxed {\left (x^{2}-5 x +6\right ) y^{\prime }+3 y x -8 y=-x^{2}} \] |
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program solution |
\[ y = \frac {-3 x^{4}+8 x^{3}+12 c_{1}}{12 x^{3}-84 x^{2}+192 x -144} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\frac {1}{4} x^{4}+\frac {2}{3} x^{3}+c_{1}}{\left (x -2\right )^{2} \left (x -3\right )} \] |
Problem 8498
| ODE |
\[ \boxed {\left (x -a \right ) \left (x -b \right ) y^{\prime }+y^{2}+k \left (y+x -a \right ) \left (y+x -b \right )=0} \] |
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program solution |
\[ y = \frac {k \left (-x +a \right )^{k +1} \left (-x +b \right )^{k +1} \left (c_{3} \left (-x +b \right )^{-k -1}+\left (-x +a \right )^{-k -1}\right )}{\left (k +1\right ) \left (c_{3} \left (-x +a \right )^{k}+\left (-x +b \right )^{k}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (\left (-x +b \right )^{1+k}+\left (-x +a \right )^{k} c_{1} \left (-x +a \right )\right ) k}{\left (1+k \right ) \left (c_{1} \left (-x +a \right )^{k}+\left (-x +b \right )^{k}\right )} \] |
Problem 8499
| ODE |
\[ \boxed {2 x^{2} y^{\prime }-2 y^{2}-y x=-2 a^{2} x} \] |
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program solution |
\[ y = \frac {\sqrt {x}\, a \left (c_{3} \cosh \left (\frac {2 a}{\sqrt {x}}\right )+\sinh \left (\frac {2 a}{\sqrt {x}}\right )\right )}{c_{3} \sinh \left (\frac {2 a}{\sqrt {x}}\right )+\cosh \left (\frac {2 a}{\sqrt {x}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tanh \left (\frac {i c_{1} \sqrt {x}+2 a}{\sqrt {x}}\right ) \sqrt {x}\, a \] |
Problem 8500
| ODE |
\[ \boxed {2 x^{2} y^{\prime }-2 y^{2}-3 y x=-2 a^{2} x} \] |
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program solution |
\[ y = \frac {\sinh \left (\frac {2 a}{\sqrt {x}}\right ) \left (2 \sqrt {x}\, a -x c_{3} \right )+2 \cosh \left (\frac {2 a}{\sqrt {x}}\right ) \left (\sqrt {x}\, c_{3} a -\frac {x}{2}\right )}{2 c_{3} \sinh \left (\frac {2 a}{\sqrt {x}}\right )+2 \cosh \left (\frac {2 a}{\sqrt {x}}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (-2 x c_{1} \sqrt {-\frac {a^{2}}{x}}-x \right ) \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )-x \left (c_{1} -2 \sqrt {-\frac {a^{2}}{x}}\right ) \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right )}{2 \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right ) c_{1} +2 \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )} \] |