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

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

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

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 0.0737839 (sec), leaf count = 27

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

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

$$\{-\frac{1}{2}\ln (\cos \left(\frac{y\left(x\right)}{x}\right))-\frac{1}{2}\ln \left(\frac{y\left(x\right)}{x}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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