##### 4.15.47 $$x y'(x) \left (x-y(x) \tan \left (\frac {y(x)}{x}\right )\right )+y(x) \left (y(x) \tan \left (\frac {y(x)}{x}\right )+x\right )=0$$

ODE
$x y'(x) \left (x-y(x) \tan \left (\frac {y(x)}{x}\right )\right )+y(x) \left (y(x) \tan \left (\frac {y(x)}{x}\right )+x\right )=0$ ODE Classiﬁcation

[[_homogeneous, class A], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.614622 (sec), leaf count = 27

$\text {Solve}\left [\log \left (\frac {y(x)}{x}\right )+\log \left (\cos \left (\frac {y(x)}{x}\right )\right )+2 \log (x)+c_1=0,y(x)\right ]$

Maple
cpu = 0.227 (sec), leaf count = 23

$\left [y \left (x \right ) = \frac {\textit {\_C1}}{\cos \left (\RootOf \left (-\textit {\_Z} \cos \left (\textit {\_Z} \right ) x^{2}+\textit {\_C1} \right )\right ) x}\right ]$ Mathematica raw input

DSolve[y[x]*(x + Tan[y[x]/x]*y[x]) + x*(x - Tan[y[x]/x]*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] + 2*Log[x] + Log[Cos[y[x]/x]] + Log[y[x]/x] == 0, y[x]]

Maple raw input

dsolve(x*(x-y(x)*tan(y(x)/x))*diff(y(x),x)+(x+y(x)*tan(y(x)/x))*y(x) = 0, y(x))

Maple raw output

[y(x) = _C1/cos(RootOf(-_Z*cos(_Z)*x^2+_C1))/x]