4.5.22 \(n y(x)+x y'(x)=f(x) g\left (x^n y(x)\right )\)

ODE
\[ n y(x)+x y'(x)=f(x) g\left (x^n y(x)\right ) \] ODE Classification

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 4.29333 (sec), leaf count = 39

\[\text {Solve}\left [\int _1^x f(K[2]) K[2]^{n-1} \, dK[2]+c_1=\int _1^{x^n y(x)} \frac {1}{g(K[1])} \, dK[1],y(x)\right ]\]

Maple
cpu = 0.272 (sec), leaf count = 32

\[ \left \{ \int \!f \relax (x ) {x}^{n-1}\,{\rm d}x-\int ^{{x}^{n}y \relax (x ) }\! \left (g \left ({\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}-{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[n*y[x] + x*y'[x] == f[x]*g[x^n*y[x]],y[x],x]

Mathematica raw output

Solve[C[1] + Integrate[f[K[2]]*K[2]^(-1 + n), {K[2], 1, x}] == Integrate[g[K[1]]
^(-1), {K[1], 1, x^n*y[x]}], y[x]]

Maple raw input

dsolve(x*diff(y(x),x)+n*y(x) = f(x)*g(x^n*y(x)), y(x),'implicit')

Maple raw output

Int(f(x)*x^(n-1),x)-Intat(1/g(_a),_a = x^n*y(x))-_C1 = 0