4.5.23 $x{y}^{\prime }(x)=y(x)f(x^{m}y(x{)}^n)$

ODE
$$x{y}^{\prime }(x)=y(x)f(x^{m}y(x{)}^n)$$ ODE Classification

[[_homogeneous, `class G`]]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 527.297 (sec), leaf count = 142

$$\text{Solve}[c_1={\int }_1^{y(x)}-\frac{K[2](nf(x^{m}K[2{]}^n)+m)({\int }_1^x\frac{m{n}^{2}K[1{]}^{m-1}K[2{]}^{n-1}{f}^{\prime }(K[1{]}^{m}K[2{]}^n)}{{(nf(K[1{]}^{m}K[2{]}^n)+m)}^2}dK[1])+n}{K[2](nf(x^{m}K[2{]}^n)+m)}dK[2]+{\int }_1^x\frac{nf(y(x{)}^{n}K[1{]}^m)}{K[1](nf(y(x{)}^{n}K[1{]}^m)+m)}dK[1],y(x)]$$

Maple ✓
cpu = 0.107 (sec), leaf count = 39

$$\{{\int }_{\mathit{\_}\mathit{b}}^{y\left(x\right)}\frac{1}{(f\left(x^m{\mathit{\_}\mathit{a}}^n\right)n+m)\mathit{\_}\mathit{a}}\mathrm{d}\mathit{\_}\mathit{a}-\frac{\ln \left(x\right)}{n}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

Int(1/(f(x^m*_a^n)*n+m)/_a,_a = _b .. y(x))-ln(x)/n-_C1 = 0