##### 4.11.18 $$x^2 \cot ^{-1}\left (\frac {y(x)}{x}\right )+x y(x) y'(x)-y(x)^2=0$$

ODE
$x^2 \cot ^{-1}\left (\frac {y(x)}{x}\right )+x y(x) y'(x)-y(x)^2=0$ ODE Classiﬁcation

[[_homogeneous, class A], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 1.02343 (sec), leaf count = 29

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

Maple
cpu = 0.228 (sec), leaf count = 21

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

DSolve[x^2*ArcCot[y[x]/x] - y[x]^2 + x*y[x]*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] == Log[x] + Inactive[Integrate][K[1]/ArcCot[K[1]], {K[1], 1, y[x]/x}]
, y[x]]

Maple raw input

dsolve(x*y(x)*diff(y(x),x)+x^2*arccot(y(x)/x)-y(x)^2 = 0, y(x))

Maple raw output

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