##### 4.20.18 $$-\left (x^2-y(x)^2\right ) y'(x)+x y(x) y'(x)^2-x y(x)=0$$

ODE
$-\left (x^2-y(x)^2\right ) y'(x)+x y(x) y'(x)^2-x y(x)=0$ ODE Classiﬁcation

[_separable]

Book solution method
No Missing Variables ODE, Solve for $$y'$$

Mathematica
cpu = 0.171273 (sec), leaf count = 45

$\left \{\left \{y(x)\to \frac {c_1}{x}\right \},\left \{y(x)\to -\sqrt {x^2+2 c_1}\right \},\left \{y(x)\to \sqrt {x^2+2 c_1}\right \}\right \}$

Maple
cpu = 0.078 (sec), leaf count = 31

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

DSolve[-(x*y[x]) - (x^2 - y[x]^2)*y'[x] + x*y[x]*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]/x}, {y[x] -> -Sqrt[x^2 + 2*C[1]]}, {y[x] -> Sqrt[x^2 + 2*C[1]]}}

Maple raw input

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

Maple raw output

[y(x) = 1/x*_C1, y(x) = (x^2+_C1)^(1/2), y(x) = -(x^2+_C1)^(1/2)]