##### 4.17.9 $$-2 a x^3 y'(x)+4 a x^2 y(x)+y'(x)^2=0$$

ODE
$-2 a x^3 y'(x)+4 a x^2 y(x)+y'(x)^2=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of variable

Mathematica
cpu = 0.834976 (sec), leaf count = 155

$\left \{\text {Solve}\left [\frac {2 \sqrt {a} x \sqrt {a x^4-4 y(x)} \tanh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {a x^4-4 y(x)}}\right )}{\sqrt {a x^2 \left (a x^4-4 y(x)\right )}}+4 c_1=\log (y(x)),y(x)\right ],\text {Solve}\left [4 c_1=\frac {2 \sqrt {a} x \sqrt {a x^4-4 y(x)} \tanh ^{-1}\left (\frac {\sqrt {a} x^2}{\sqrt {a x^4-4 y(x)}}\right )}{\sqrt {a x^2 \left (a x^4-4 y(x)\right )}}+\log (y(x)),y(x)\right ]\right \}$

Maple
cpu = 3.365 (sec), leaf count = 27

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

DSolve[4*a*x^2*y[x] - 2*a*x^3*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[4*C[1] + (2*Sqrt[a]*x*ArcTanh[(Sqrt[a]*x^2)/Sqrt[a*x^4 - 4*y[x]]]*Sqrt[a*
x^4 - 4*y[x]])/Sqrt[a*x^2*(a*x^4 - 4*y[x])] == Log[y[x]], y[x]], Solve[4*C[1] ==
 Log[y[x]] + (2*Sqrt[a]*x*ArcTanh[(Sqrt[a]*x^2)/Sqrt[a*x^4 - 4*y[x]]]*Sqrt[a*x^4
 - 4*y[x]])/Sqrt[a*x^2*(a*x^4 - 4*y[x])], y[x]]}

Maple raw input

dsolve(diff(y(x),x)^2-2*a*x^3*diff(y(x),x)+4*a*x^2*y(x) = 0, y(x))

Maple raw output

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