# Liouville normal form

$\newcommand{deriv}{\frac{\mathrm{d}#1}{\mathrm{d}#2}} \newcommand{derivn}{\frac{\mathrm{d}^{#3}#1}{\mathrm{d}#2^{#3}}}$
The Liouville normal form is a way of writing a second-order ordinary linear differential equation \begin{equation}\label{eq1} \derivn{y}{x}{2} + p(x)\deriv{y}{x} + \left( q(x) + \lambda r(x) \right) y = 0, \end{equation} in the form \begin{equation}\label{eq2} \derivn{\eta}{\xi}{2} + \left( \lambda + \phi(\xi) \right) \eta = 0, \end{equation} where $\lambda$ is parameter. If $p(x) \in C^1$, $r(x) \in C^2$ and $r(x) > 0$, then equation \ref{eq1} reduces to the Liouville normal form \ref{eq2} by means of the substitution $\eta(\xi) = \Phi(x)y(x),\quad \xi = \int_\alpha^x \sqrt{r(t)}\,\mathrm{d}t, \quad \Phi(x) = r(x)^{1/4} \exp\left( \frac{1}{2}\int_\alpha^x p(t)\,\mathrm{d}t \right),$ which is called the Liouville transformation (introduced in [Li]). The Liouville normal form plays an important role in the investigation of the asymptotic behaviour of solutions of \ref{eq1} for large values of the parameter $\lambda$ or the argument, and in the investigation of the asymptotics of eigenfunctions and eigenvalues of the Sturm–Liouville problem (see [Ti]).