4.24.21 $x^{n}f(y^{\prime }(x),\frac{y(x)}{x})=0$

ODE
$$x^{n}f(y^{\prime }(x),\frac{y(x)}{x})=0$$ ODE Classification

[[_homogeneous, `class A`], _dAlembert]

Book solution method
Homogeneous ODE, $x^{n}f(\frac{y}{x},y^{\prime })=0$

Mathematica ✓
cpu = 0.877079 (sec), leaf count = 34

$$\text{Solve}[c_1={\int }_1^{\frac{y(x)}{x}}\frac{1}{K[1]-\text{InverseFunction}[f,1,2][0,K[1]]}dK[1]+\log (x),y(x)]$$

Maple ✓
cpu = 0.033 (sec), leaf count = 30

$$\{\ln \left(x\right)-{\int }^{\frac{y\left(x\right)}{x}}{(\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}\left(f(\mathit{\_}\mathit{Z},\mathit{\_}\mathit{a})\right)-\mathit{\_}\mathit{a})}^{-1}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[x^n*f[y'[x], y[x]/x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] == Integrate[(K[1] - InverseFunction[f, 1, 2][0, K[1]])^(-1), {K[1], 
1, y[x]/x}] + Log[x], y[x]]

Maple raw input

dsolve(x^n*f(diff(y(x),x),y(x)/x) = 0, y(x),'implicit')

Maple raw output

ln(x)-Intat(1/(RootOf(f(_Z,_a))-_a),_a = y(x)/x)-_C1 = 0