Reducing order of a Non Linear Second Order ODE Let be $u=u(x)$ so:
$$u''=f(u) \left[ (u')^2+g(x)u'\right]$$
Where the apex denotes x-derivative. 
In my case: $f(u)=-1/u$ and $g(x)=x/2$.
Is there a change of variable in order to reduce the order of ODE?
 A: The differential equation does permit a scaling symmetry of $x$ and $u$. Note, that the differential equation is invariant under the transformation
$$\tilde{x}=\exp(\varepsilon)x$$
$$\tilde{u}=\exp(2\varepsilon)u.$$
You can then try the method of canonical coordinates to obtain the substitutions:
$$s(x,u)=\ln x \implies x(r,s) = \exp(s)$$
$$r(x,u)=u/x^2 \implies u(r,s) = r\exp(2s).$$ 
The resulting ODE is given by (calculated with Maple):
$$-{\frac {-2\, \left( {\frac {\rm d}{{\rm d}r}}s \left( r \right) 
 \right) ^{3}r-3\, \left( {\frac {\rm d}{{\rm d}r}}s \left( r \right) 
 \right) ^{2}+{\frac {{\rm d}^{2}}{{\rm d}{r}^{2}}}s \left( r \right) 
}{ \left( {\frac {\rm d}{{\rm d}r}}s \left( r \right)  \right) ^{3}}}=
-1/2\,{\frac { \left( 2\,r{\frac {\rm d}{{\rm d}r}}s \left( r \right) 
+1 \right)  \left( 4\,r{\frac {\rm d}{{\rm d}r}}s \left( r \right) +{
\frac {\rm d}{{\rm d}r}}s \left( r \right) +2 \right) }{r \left( {
\frac {\rm d}{{\rm d}r}}s \left( r \right)  \right) ^{2}}}
$$
Note that the differential can be expressed as a nonlinear first order ODE by the substitution $s'(r)=k(r)$ and you can further simplify this.
