4.38.42 $f(x{)}^2{y}^{″}(x)=-af(x{)}^5+3f(x)f^{\prime }(x)-f(x{)}^{2}y(x)+3f(x{)}^3$

ODE
$$f(x{)}^2{y}^{″}(x)=-af(x{)}^5+3f(x)f^{\prime }(x)-f(x{)}^{2}y(x)+3f(x{)}^3$$ ODE Classification

(ODEtools/info) missing specification of intermediate function

Book solution method
TO DO

Mathematica ✓
cpu = 1.21559 (sec), leaf count = 97

$$\left\{\{y(x)\to \cos (x)({\int }_1^x\frac{\sin (K[1])(af(K[1]{)}^4-3f^{\prime }(K[1])-3f(K[1]{)}^2)}{f(K[1])}dK[1])+\sin (x)({\int }_1^x-\frac{\cos (K[2])(af(K[2]{)}^4-3f^{\prime }(K[2])-3f(K[2]{)}^2)}{f(K[2])}dK[2])+c_2\sin (x)+c_1\cos (x)\}\right\}$$

Maple ✓
cpu = 0.129 (sec), leaf count = 76

$$\{y\left(x\right)=\sin \left(x\right)\mathit{\_}\mathit{C}\mathit{2}+\cos \left(x\right)\mathit{\_}\mathit{C}\mathit{1}-\int \frac{\cos \left(x\right)({\left(f\left(x\right)\right)}^{4}a-3{\left(f\left(x\right)\right)}^2-3\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right))}{f\left(x\right)}\mathrm{d}x\sin \left(x\right)+\int \frac{\sin \left(x\right)({\left(f\left(x\right)\right)}^{4}a-3{\left(f\left(x\right)\right)}^2-3\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right))}{f\left(x\right)}\mathrm{d}x\cos \left(x\right)\}$$ 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)