##### 4.19.15 $$x^2 y'(x)^2+(y(x)+2 x) y(x) y'(x)+y(x)^2=0$$

ODE
$x^2 y'(x)^2+(y(x)+2 x) y(x) y'(x)+y(x)^2=0$ ODE Classiﬁcation

[[_homogeneous, class A], _dAlembert]

Book solution method
Change of variable

Mathematica
cpu = 0.364661 (sec), leaf count = 64

$\left \{\left \{y(x)\to -\frac {\cosh (4 c_1)+\sinh (4 c_1)}{-x+\cosh (2 c_1)+\sinh (2 c_1)}\right \},\left \{y(x)\to \frac {\cosh (4 c_1)+\sinh (4 c_1)}{x+\cosh (2 c_1)+\sinh (2 c_1)}\right \}\right \}$

Maple
cpu = 7.292 (sec), leaf count = 121

$\left [y \left (x \right ) = -4 x, y \left (x \right ) = \frac {2 \textit {\_C1}^{2} \left (\sqrt {2}\, \textit {\_C1} -x \right )}{2 \textit {\_C1}^{2}-x^{2}}, y \left (x \right ) = -\frac {2 \textit {\_C1}^{2} \left (\sqrt {2}\, \textit {\_C1} +x \right )}{2 \textit {\_C1}^{2}-x^{2}}, y \left (x \right ) = \frac {\textit {\_C1}^{2} \left (\sqrt {2}\, \textit {\_C1} -2 x \right )}{2 \textit {\_C1}^{2}-4 x^{2}}, y \left (x \right ) = -\frac {\textit {\_C1}^{2} \left (\sqrt {2}\, \textit {\_C1} +2 x \right )}{2 \left (\textit {\_C1}^{2}-2 x^{2}\right )}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((Cosh[4*C[1]] + Sinh[4*C[1]])/(-x + Cosh[2*C[1]] + Sinh[2*C[1]]))},
{y[x] -> (Cosh[4*C[1]] + Sinh[4*C[1]])/(x + Cosh[2*C[1]] + Sinh[2*C[1]])}}

Maple raw input

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

Maple raw output

[y(x) = -4*x, y(x) = 2*_C1^2*(2^(1/2)*_C1-x)/(2*_C1^2-x^2), y(x) = -2*_C1^2*(2^(
1/2)*_C1+x)/(2*_C1^2-x^2), y(x) = 1/2*_C1^2*(2^(1/2)*_C1-2*x)/(_C1^2-2*x^2), y(x
) = -1/2*_C1^2*(2^(1/2)*_C1+2*x)/(_C1^2-2*x^2)]