ODE
\[ a x^3 y'(x)-2 a x^2 y(x)+y'(x)^2=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries]]
Book solution method
Homogeneous ODE, The Isobaric equation
Mathematica ✓
cpu = 0.363012 (sec), leaf count = 98
\[\left \{\left \{y(x)\to \frac {1}{2} (\cosh (2 c_1)+\sinh (2 c_1)) \left (-\sqrt {2} \sqrt {a} x^2+2 \cosh (2 c_1)+2 \sinh (2 c_1)\right )\right \},\left \{y(x)\to \frac {1}{2} (\cosh (2 c_1)+\sinh (2 c_1)) \left (\sqrt {2} \sqrt {a} x^2+2 \cosh (2 c_1)+2 \sinh (2 c_1)\right )\right \}\right \}\]
Maple ✓
cpu = 3.324 (sec), leaf count = 27
\[\left [y \left (x \right ) = -\frac {a \,x^{4}}{8}, y \left (x \right ) = x^{2} \textit {\_C1} +\frac {2 \textit {\_C1}^{2}}{a}\right ]\] Mathematica raw input
DSolve[-2*a*x^2*y[x] + a*x^3*y'[x] + y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((Cosh[2*C[1]] + Sinh[2*C[1]])*(-(Sqrt[2]*Sqrt[a]*x^2) + 2*Cosh[2*C[1]] + 2*Sinh[2*C[1]]))/2}, {y[x] -> ((Cosh[2*C[1]] + Sinh[2*C[1]])*(Sqrt[2]*Sqrt[a]*x^2 + 2*Cosh[2*C[1]] + 2*Sinh[2*C[1]]))/2}}
Maple raw input
dsolve(diff(y(x),x)^2+a*x^3*diff(y(x),x)-2*a*x^2*y(x) = 0, y(x))
Maple raw output
[y(x) = -1/8*a*x^4, y(x) = x^2*_C1+2*_C1^2/a]