4.17.8 \(a x^3 y'(x)-2 a x^2 y(x)+y'(x)^2=0\)

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 = 0.219418 (sec), leaf count = 56

\[\left \{\left \{y(x)\to 2 a e^{2 c_1} \left (4 e^{2 c_1}+x^2\right )\right \},\left \{y(x)\to \frac {e^{4 c_1}-2 a e^{2 c_1} x^2}{8 a}\right \}\right \}\]

Maple
cpu = 0.413 (sec), leaf count = 27

\[ \left \{ y \relax (x ) =-{\frac {a{x}^{4}}{8}},y \relax (x ) ={\it \_C1}\,{x}^{2}+2\,{\frac {{{\it \_C1}}^{2}}{a}} \right \} \] 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] -> 2*a*E^(2*C[1])*(4*E^(2*C[1]) + x^2)}, {y[x] -> (E^(4*C[1]) - 2*a*E^(2*
C[1])*x^2)/(8*a)}}

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),'implicit')

Maple raw output

y(x) = -1/8*a*x^4, y(x) = _C1*x^2+2*_C1^2/a