##### 4.11.29 $$x (y(x)+x) y'(x)+y(x)^2=0$$

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

[[_homogeneous, class A], _rational, [_Abel, 2nd type, class B]]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.300084 (sec), leaf count = 75

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

Maple
cpu = 0.174 (sec), leaf count = 44

$\left [y \left (x \right ) = \frac {1+\sqrt {x^{2} \textit {\_C1} +1}}{\textit {\_C1} x}, y \left (x \right ) = -\frac {-1+\sqrt {x^{2} \textit {\_C1} +1}}{\textit {\_C1} x}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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