3.30 ODE change of variable. Dependent variable

given an ode

\[ y \left (x \right ) = \left (\frac {d}{d x}y \left (x \right )\right )^{3} y \left (x \right )^{2}+2 x \left (\frac {d}{d x}y \left (x \right )\right ) \]

do change of variable \(u(x)=y(x)^2\)

restart; 
ode:=y(x)=diff(y(x),x)^3*y(x)^2+2*x*diff(y(x),x); 
new_ode:=PDEtools:-dchange({y(x)=sqrt(u(x))},ode,{u});
 

\[ \sqrt {u \left (x \right )} = \frac {\left (\frac {d}{d x}u \left (x \right )\right )^{3}}{8 \sqrt {u \left (x \right )}}+\frac {x \left (\frac {d}{d x}u \left (x \right )\right )}{\sqrt {u \left (x \right )}} \]