ODE
\[ f(x)^2 y''(x)=-a f(x)^5+3 f(x) f'(x)-f(x)^2 y(x)+3 f(x)^3 \] ODE Classification
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Book solution method
TO DO
Mathematica ✓
cpu = 0.857136 (sec), leaf count = 99
\[\left \{\left \{y(x)\to \cos (x) \int _1^x\frac {\sin (K[1]) \left (a f(K[1])^4-3 f(K[1])^2-3 f'(K[1])\right )}{f(K[1])}dK[1]+\sin (x) \int _1^x-\frac {\cos (K[2]) \left (a f(K[2])^4-3 f(K[2])^2-3 f'(K[2])\right )}{f(K[2])}dK[2]+c_1 \cos (x)+c_2 \sin (x)\right \}\right \}\]
Maple ✓
cpu = 0.24 (sec), leaf count = 76
\[\left [y \left (x \right ) = \sin \left (x \right ) \textit {\_C2} +\textit {\_C1} \cos \left (x \right )-\left (\int \frac {\cos \left (x \right ) \left (f \left (x \right )^{4} a -3 f \left (x \right )^{2}-3 \frac {d}{d x}f \left (x \right )\right )}{f \left (x \right )}d x \right ) \sin \left (x \right )+\left (\int \frac {\sin \left (x \right ) \left (f \left (x \right )^{4} a -3 f \left (x \right )^{2}-3 \frac {d}{d x}f \left (x \right )\right )}{f \left (x \right )}d x \right ) \cos \left (x \right )\right ]\] Mathematica raw input
DSolve[f[x]^2*y''[x] == 3*f[x]^3 - a*f[x]^5 - f[x]^2*y[x] + 3*f[x]*f'[x],y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cos[x] + C[2]*Sin[x] + Cos[x]*Inactive[Integrate][(Sin[K[1]]*(-3*f[K[1]]^2 + a*f[K[1]]^4 - 3*Derivative[1][f][K[1]]))/f[K[1]], {K[1], 1, x}] + Sin[x]*Inactive[Integrate][-((Cos[K[2]]*(-3*f[K[2]]^2 + a*f[K[2]]^4 - 3*Derivative[1][f][K[2]]))/f[K[2]]), {K[2], 1, x}]}}
Maple raw input
dsolve(f(x)^2*diff(diff(y(x),x),x) = 3*f(x)^3+3*f(x)*diff(f(x),x)-f(x)^2*y(x)-a*f(x)^5, y(x))
Maple raw output
[y(x) = sin(x)*_C2+_C1*cos(x)-Int(cos(x)/f(x)*(f(x)^4*a-3*f(x)^2-3*diff(f(x),x)),x)*sin(x)+Int(sin(x)/f(x)*(f(x)^4*a-3*f(x)^2-3*diff(f(x),x)),x)*cos(x)]