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]