ODE
\[ y''(x)=y(x) \left (f'(x)-2 f(x)^2\right )+(3 f(x)-y(x)) y'(x)+f(x) y(x)^2+y(x)^3 \] ODE Classification
odeadvisor timed out
Book solution method
TO DO
Mathematica ✗
cpu = 0.363712 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == f[x]*y[x]^2 + y[x]^3 + y[x]*(-2*f[x]^2 + Derivative[1][f][x]) + (3*f[x] - y[x])*Derivative[1][y][x], y[x], x]
Maple ✓
cpu = 2.264 (sec), leaf count = 828
\[\left [y \left (x \right ) = 0, y \left (x \right ) = \frac {{\mathrm e}^{\int f \left (x \right )d x}}{\int {\mathrm e}^{\int f \left (x \right )d x}d x +\textit {\_C3}}, y \left (x \right ) = -\frac {2 \,{\mathrm e}^{\int f \left (x \right )d x}}{\int {\mathrm e}^{\int f \left (x \right )d x}d x +\textit {\_C3}}, y \left (x \right ) = \RootOf \left (-\left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right )+\int _{}^{\textit {\_Z}}\frac {\textit {\_f}^{8}-\textit {\_C1} \,\textit {\_f}^{2}+\left (-\textit {\_f}^{12}+2 \textit {\_C1} \,\textit {\_f}^{6}-\textit {\_C1}^{2}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1} \,\textit {\_f}^{6}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1}^{2}\right )^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+\textit {\_C1} \right ) \left (-\textit {\_f}^{12}+2 \textit {\_C1} \,\textit {\_f}^{6}-\textit {\_C1}^{2}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1} \,\textit {\_f}^{6}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1}^{2}\right )^{\frac {1}{3}}}d \textit {\_f} +\textit {\_C2} \right ) {\mathrm e}^{\int f \left (x \right )d x}, y \left (x \right ) = \RootOf \left (-2 \left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right )-\left (\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}+\textit {\_f}^{8}+i \sqrt {3}\, \textit {\_C1} \,\textit {\_f}^{2}+i \sqrt {3}\, \left (-\textit {\_f}^{12}+2 \textit {\_C1} \,\textit {\_f}^{6}-\textit {\_C1}^{2}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1} \,\textit {\_f}^{6}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1}^{2}\right )^{\frac {2}{3}}-\textit {\_C1} \,\textit {\_f}^{2}+\left (-\textit {\_f}^{12}+2 \textit {\_C1} \,\textit {\_f}^{6}-\textit {\_C1}^{2}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1} \,\textit {\_f}^{6}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1}^{2}\right )^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+\textit {\_C1} \right ) \left (-\textit {\_f}^{12}+2 \textit {\_C1} \,\textit {\_f}^{6}-\textit {\_C1}^{2}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1} \,\textit {\_f}^{6}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1}^{2}\right )^{\frac {1}{3}}}d \textit {\_f} \right )+2 \textit {\_C2} \right ) {\mathrm e}^{\int f \left (x \right )d x}, y \left (x \right ) = \RootOf \left (-2 \left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right )+\int _{}^{\textit {\_Z}}\frac {-i \sqrt {3}\, \textit {\_f}^{8}-\textit {\_f}^{8}+i \sqrt {3}\, \textit {\_C1} \,\textit {\_f}^{2}+i \sqrt {3}\, \left (-\textit {\_f}^{12}+2 \textit {\_C1} \,\textit {\_f}^{6}-\textit {\_C1}^{2}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1} \,\textit {\_f}^{6}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1}^{2}\right )^{\frac {2}{3}}+\textit {\_C1} \,\textit {\_f}^{2}-\left (-\textit {\_f}^{12}+2 \textit {\_C1} \,\textit {\_f}^{6}-\textit {\_C1}^{2}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1} \,\textit {\_f}^{6}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1}^{2}\right )^{\frac {2}{3}}}{\left (-\textit {\_f}^{6}+\textit {\_C1} \right ) \left (-\textit {\_f}^{12}+2 \textit {\_C1} \,\textit {\_f}^{6}-\textit {\_C1}^{2}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_f}^{12}-2 \sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1} \,\textit {\_f}^{6}+\sqrt {\frac {\textit {\_C1}}{-\textit {\_f}^{6}+\textit {\_C1}}}\, \textit {\_C1}^{2}\right )^{\frac {1}{3}}}d \textit {\_f} +2 \textit {\_C2} \right ) {\mathrm e}^{\int f \left (x \right )d x}\right ]\] Mathematica raw input
DSolve[y''[x] == f[x]*y[x]^2 + y[x]^3 + y[x]*(-2*f[x]^2 + f'[x]) + (3*f[x] - y[x])*y'[x],y[x],x]
Mathematica raw output
DSolve[Derivative[2][y][x] == f[x]*y[x]^2 + y[x]^3 + y[x]*(-2*f[x]^2 + Derivative[1][f][x]) + (3*f[x] - y[x])*Derivative[1][y][x], y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x) = (3*f(x)-y(x))*diff(y(x),x)+(diff(f(x),x)-2*f(x)^2)*y(x)+f(x)*y(x)^2+y(x)^3, y(x))
Maple raw output
[y(x) = 0, y(x) = exp(Int(f(x),x))/(Int(exp(Int(f(x),x)),x)+_C3), y(x) = -2*exp(Int(f(x),x))/(Int(exp(Int(f(x),x)),x)+_C3), y(x) = RootOf(-Int(exp(Int(f(x),x)),x)+Intat((_f^8-_C1*_f^2+(-_f^12+2*_C1*_f^6-_C1^2+(_C1/(-_f^6+_C1))^(1/2)*_f^12-2*(_C1/(-_f^6+_C1))^(1/2)*_C1*_f^6+(_C1/(-_f^6+_C1))^(1/2)*_C1^2)^(2/3))/(-_f^6+_C1)/(-_f^12+2*_C1*_f^6-_C1^2+(_C1/(-_f^6+_C1))^(1/2)*_f^12-2*(_C1/(-_f^6+_C1))^(1/2)*_C1*_f^6+(_C1/(-_f^6+_C1))^(1/2)*_C1^2)^(1/3),_f = _Z)+_C2)/exp(-Int(f(x),x)), y(x) = RootOf(-2*Int(exp(Int(f(x),x)),x)-Intat((-I*3^(1/2)*_f^8+_f^8+I*3^(1/2)*_C1*_f^2+I*3^(1/2)*(-_f^12+2*_C1*_f^6-_C1^2+(_C1/(-_f^6+_C1))^(1/2)*_f^12-2*(_C1/(-_f^6+_C1))^(1/2)*_C1*_f^6+(_C1/(-_f^6+_C1))^(1/2)*_C1^2)^(2/3)-_C1*_f^2+(-_f^12+2*_C1*_f^6-_C1^2+(_C1/(-_f^6+_C1))^(1/2)*_f^12-2*(_C1/(-_f^6+_C1))^(1/2)*_C1*_f^6+(_C1/(-_f^6+_C1))^(1/2)*_C1^2)^(2/3))/(-_f^6+_C1)/(-_f^12+2*_C1*_f^6-_C1^2+(_C1/(-_f^6+_C1))^(1/2)*_f^12-2*(_C1/(-_f^6+_C1))^(1/2)*_C1*_f^6+(_C1/(-_f^6+_C1))^(1/2)*_C1^2)^(1/3),_f = _Z)+2*_C2)/exp(-Int(f(x),x)), y(x) = RootOf(-2*Int(exp(Int(f(x),x)),x)+Intat((-I*3^(1/2)*_f^8-_f^8+I*3^(1/2)*_C1*_f^2+I*3^(1/2)*(-_f^12+2*_C1*_f^6-_C1^2+(_C1/(-_f^6+_C1))^(1/2)*_f^12-2*(_C1/(-_f^6+_C1))^(1/2)*_C1*_f^6+(_C1/(-_f^6+_C1))^(1/2)*_C1^2)^(2/3)+_C1*_f^2-(-_f^12+2*_C1*_f^6-_C1^2+(_C1/(-_f^6+_C1))^(1/2)*_f^12-2*(_C1/(-_f^6+_C1))^(1/2)*_C1*_f^6+(_C1/(-_f^6+_C1))^(1/2)*_C1^2)^(2/3))/(-_f^6+_C1)/(-_f^12+2*_C1*_f^6-_C1^2+(_C1/(-_f^6+_C1))^(1/2)*_f^12-2*(_C1/(-_f^6+_C1))^(1/2)*_C1*_f^6+(_C1/(-_f^6+_C1))^(1/2)*_C1^2)^(1/3),_f = _Z)+2*_C2)/exp(-Int(f(x),x))]