4.19.39 \(x^4 y'(x)^2+x y(x)^2 y'(x)-y(x)^3=0\)

ODE
\[ x^4 y'(x)^2+x y(x)^2 y'(x)-y(x)^3=0 \] ODE Classification

[[_homogeneous, `class G`]]

Book solution method
Change of variable

Mathematica
cpu = 0.20814 (sec), leaf count = 50

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

Maple
cpu = 0.177 (sec), leaf count = 84

\[ \left \{ 2\, \left (y \relax (x ) \right ) ^{-1}+{x}^{-2}+{\frac {1}{{x}^{2}y \relax (x ) }\sqrt { \left (y \relax (x ) \right ) ^{2}+4\,{x}^{2}y \relax (x ) }}-{\it \_C1}=0,2\,y \relax (x ) +{\frac { \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}+{\frac {y \relax (x ) }{{x}^{2}}\sqrt { \left (y \relax (x ) \right ) ^{2}+4\,{x}^{2}y \relax (x ) }}-{\it \_C1}=0,y \relax (x ) =-4\,{x}^{2} \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-4*E^(2*C[1])*x)/(2*E^C[1] - x)}, {y[x] -> (E^(2*C[1])*x)/(2*E^C[1] +
 4*x)}}

Maple raw input

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

Maple raw output

y(x) = -4*x^2, 2*y(x)+y(x)^2/x^2+y(x)/x^2*(y(x)^2+4*x^2*y(x))^(1/2)-_C1 = 0, 2/y
(x)+1/x^2+1/y(x)/x^2*(y(x)^2+4*x^2*y(x))^(1/2)-_C1 = 0