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 = 0.0737839 (sec), leaf count = 27

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

Maple
cpu = 0.021 (sec), leaf count = 30

\[ \left \{ -{\frac {1}{2}\ln \left (\cos \left ({\frac {y \relax (x ) }{x}} \right ) \right ) }-{\frac {1}{2}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

-1/2*ln(cos(y(x)/x))-1/2*ln(y(x)/x)-ln(x)-_C1 = 0