Nonlinear ODE $a^2u''+bu'+b^*(u')^2/u+cu=-1$

Question

Consider the ODE $$[a(x)]^2u''+b(x)u'+b^*(x)(u')^2/u+c(x)u=-1$$ on $\mathbb{R}$. I am trying to make it looks better by writing it as a variational problem ($u$ is some minimizer of some energy functionals). Do anyone have ideas about that? Or any good change of variable to reduce the nonlinearity of the equation? My only thought is to write $u=e^v$ and $$[a(x)]^2v''+b(x)v'+\{b^*(x)+[a(x)]^2\}(v')^2+c(x)=-e^{-v},$$ but I do not know if it is useful.