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
(ODEtools/info) missing specification of intermediate function
Book solution method
TO DO
Mathematica ✗
cpu = 0.455828 (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 = 0.783 (sec), leaf count = 442
\[ \left \{ \int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int ^{y \left ( x \right ) {{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}}\!{\frac {1}{2\,{{\it \_f}}^{6}-2\,{\it \_C1}} \left ( -{{{\it \_f}}^{2} \left ( i\sqrt {3}-1 \right ) \left ( {{\it \_f}}^{6}-{\it \_C1} \right ) {\frac {1}{\sqrt [3]{ \left ( {{\it \_f}}^{6}-{\it \_C1} \right ) ^{2} \left ( -1+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}} \right ) }}}}+\sqrt [3]{ \left ( {{\it \_f}}^{6}-{\it \_C1} \right ) ^{2} \left ( -1+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}} \right ) } \left ( i\sqrt {3}+1 \right ) \right ) }{d{\it \_f}}-{\it \_C2}=0,\int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int ^{y \left ( x \right ) {{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}}\!{\frac {1}{2\,{{\it \_f}}^{6}-2\,{\it \_C1}} \left ( {{{\it \_f}}^{2} \left ( i\sqrt {3}+1 \right ) \left ( {{\it \_f}}^{6}-{\it \_C1} \right ) {\frac {1}{\sqrt [3]{ \left ( {{\it \_f}}^{6}-{\it \_C1} \right ) ^{2} \left ( -1+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}} \right ) }}}}+ \left ( -i\sqrt {3}+1 \right ) \sqrt [3]{ \left ( {{\it \_f}}^{6}-{\it \_C1} \right ) ^{2} \left ( -1+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}} \right ) } \right ) }{d{\it \_f}}-{\it \_C2}=0,\int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x-\int ^{y \left ( x \right ) {{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}}\!{\frac {1}{-{{\it \_f}}^{6}+{\it \_C1}}\sqrt [3]{ \left ( {{\it \_f}}^{6}-{\it \_C1} \right ) ^{2} \left ( -1+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}} \right ) }}-{{{\it \_f}}^{2}{\frac {1}{\sqrt [3]{ \left ( {{\it \_f}}^{6}-{\it \_C1} \right ) ^{2} \left ( -1+\sqrt {{\frac {{\it \_C1}}{-{{\it \_f}}^{6}+{\it \_C1}}}} \right ) }}}}{d{\it \_f}}-{\it \_C2}=0,y \left ( x \right ) =0,y \left ( x \right ) ={\frac {1}{2\,{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}} \left ( \int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{\it \_C3} \right ) } \left ( -1-\sqrt {5+4\,{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}} \right ) },y \left ( x \right ) ={\frac {1}{2\,{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}} \left ( \int \!{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}\,{\rm d}x+{\it \_C3} \right ) } \left ( -1+\sqrt {5+4\,{{\rm e}^{\int \!f \left ( x \right ) \,{\rm d}x}}{{\rm e}^{-\int \!f \left ( x \right ) \,{\rm d}x}}} \right ) } \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),'implicit')
Maple raw output
y(x) = 0, y(x) = 1/2*(-1-(5+4*exp(Int(f(x),x))*exp(-Int(f(x),x)))^(1/2))/exp(-Int(f(x),x))/(Int(exp(Int(f(x),x)),x)+_C3), y(x) = 1/2*(-1+(5+4*exp(Int(f(x),x))*exp(-Int(f(x),x)))^(1/2))/exp(-Int(f(x),x))/(Int(exp(Int(f(x),x)),x)+_C3), Int(exp(Int(f(x),x)),x)-Intat(1/(-_f^6+_C1)*((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3)-_f^2/((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3),_f = y(x)*exp(-Int(f(x),x)))-_C2 = 0, Int(exp(Int(f(x),x)),x)-Intat((-_f^2*(I*3^(1/2)-1)*(_f^6-_C1)/((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3)+((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3)*(I*3^(1/2)+1))/(2*_f^6-2*_C1),_f = y(x)*exp(-Int(f(x),x)))-_C2 = 0, Int(exp(Int(f(x),x)),x)-Intat((_f^2*(I*3^(1/2)+1)*(_f^6-_C1)/((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3)+(-I*3^(1/2)+1)*((_f^6-_C1)^2*(-1+(_C1/(-_f^6+_C1))^(1/2)))^(1/3))/(2*_f^6-2*_C1),_f = y(x)*exp(-Int(f(x),x)))-_C2 = 0