##### 4.12.30 $$x (2 x y(x)+1) y'(x)+y(x) (3 x y(x)+2)=0$$

ODE
$x (2 x y(x)+1) y'(x)+y(x) (3 x y(x)+2)=0$ ODE Classiﬁcation

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

Book solution method
Homogeneous equation, isobaric equation

Mathematica
cpu = 0.500822 (sec), leaf count = 69

$\left \{\left \{y(x)\to -\frac {x^{3/2}+\sqrt {x^2 (x+4 c_1)}}{2 x^{5/2}}\right \},\left \{y(x)\to \frac {-x^{3/2}+\sqrt {x^2 (x+4 c_1)}}{2 x^{5/2}}\right \}\right \}$

Maple
cpu = 0.068 (sec), leaf count = 43

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

DSolve[y[x]*(2 + 3*x*y[x]) + x*(1 + 2*x*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -1/2*(x^(3/2) + Sqrt[x^2*(x + 4*C[1])])/x^(5/2)}, {y[x] -> (-x^(3/2) +
 Sqrt[x^2*(x + 4*C[1])])/(2*x^(5/2))}}

Maple raw input

dsolve(x*(1+2*x*y(x))*diff(y(x),x)+(2+3*x*y(x))*y(x) = 0, y(x))

Maple raw output

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