4.19.33 $$a+x^3 y'(x)^2+x^2 y(x) y'(x)=0$$

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

[[_homogeneous, class G], _rational]

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

Mathematica
cpu = 1.17885 (sec), leaf count = 123

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

Maple
cpu = 3.163 (sec), leaf count = 66

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

DSolve[a + x^2*y[x]*y'[x] + x^3*y'[x]^2 == 0,y[x],x]

Mathematica raw output

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

Maple raw input

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

Maple raw output

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