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
(ODEtools/info) missing specification of intermediate function
Book solution method
TO DO
Mathematica ✓
cpu = 1.21559 (sec), leaf count = 97
\[\left \{\left \{y(x)\to \cos (x) \left (\int _1^x \frac {\sin (K[1]) \left (a f(K[1])^4-3 f'(K[1])-3 f(K[1])^2\right )}{f(K[1])} \, dK[1]\right )+\sin (x) \left (\int _1^x -\frac {\cos (K[2]) \left (a f(K[2])^4-3 f'(K[2])-3 f(K[2])^2\right )}{f(K[2])} \, dK[2]\right )+c_2 \sin (x)+c_1 \cos (x)\right \}\right \}\]
Maple ✓
cpu = 0.129 (sec), leaf count = 76
\[ \left \{ y \left ( x \right ) =\sin \left ( x \right ) {\it \_C2}+\cos \left ( x \right ) {\it \_C1}-\int \!{\frac {\cos \left ( x \right ) \left ( \left ( f \left ( x \right ) \right ) ^{4}a-3\, \left ( f \left ( x \right ) \right ) ^{2}-3\,{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) }{f \left ( x \right ) }}\,{\rm d}x\sin \left ( x \right ) +\int \!{\frac {\sin \left ( x \right ) \left ( \left ( f \left ( x \right ) \right ) ^{4}a-3\, \left ( f \left ( x \right ) \right ) ^{2}-3\,{\frac {\rm d}{{\rm d}x}}f \left ( x \right ) \right ) }{f \left ( x \right ) }}\,{\rm d}x\cos \left ( x \right ) \right \} \] 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] + Cos[x]*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}] + C[2]*Sin[x] + 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}]*Sin[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),'implicit')
Maple raw output
y(x) = sin(x)*_C2+cos(x)*_C1-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)