View source for Liouville normal form

{{MSC|34A30|34B24}}
{{TEX|done}}

$
\newcommand{deriv}[2]{\frac{\mathrm{d}#1}{\mathrm{d}#2}}
\newcommand{derivn}[3]{\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 {{Cite|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 {{Cite|Ti}}).

====References==== 
{|
|-
|valign="top"|{{Ref|In}}||valign="top"| E.L. Ince, "Ordinary differential equations", Dover, reprint (1956)
|-
|valign="top"|{{Ref|Ka}}||valign="top"| E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", '''1. Gewöhnliche Differentialgleichungen''', Chelsea, reprint (1947)
|-
|valign="top"|{{Ref|Li}}||valign="top"| J. Liouville, ''J. Math. Pures Appl.'', '''2''' (1837) pp. 16–35
|-
|valign="top"|{{Ref|Ti}}||valign="top"| E.C. Titchmarsh, "Eigenfunction expansions associated with second-order differential equations", '''1–2''', Clarendon Press (1946–1948)
|-
|}