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]