##### 4.24.21 $$x^n f\left (y'(x),\frac {y(x)}{x}\right )=0$$

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

[[_homogeneous, class A], _dAlembert]

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

Mathematica
cpu = 0.187663 (sec), leaf count = 35

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

Maple
cpu = 0.168 (sec), leaf count = 27

$\left [y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {1}{\RootOf \left (f \left (\textit {\_Z} , \textit {\_a}\right )\right )-\textit {\_a}}d \textit {\_a} +\textit {\_C1} \right ) x\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = RootOf(-ln(x)+Intat(1/(RootOf(f(_Z,_a))-_a),_a = _Z)+_C1)*x]