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A way of writting a second-order ordinary linear differential equation
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{{MSC|34A30|34B24}}
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$
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\newcommand{deriv}[2]{\frac{\mathrm{d}#1}{\mathrm{d}#2}}
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\newcommand{derivn}[3]{\frac{\mathrm{d}^{#3}#1}{\mathrm{d}#2^{#3}}}
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$
  
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The Liouville normal form is a way of writing a second-order ordinary linear differential equation
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\begin{equation}\label{eq1}
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\derivn{y}{x}{2} +
 +
p(x)\deriv{y}{x} +
 +
\left(
 +
q(x) + \lambda r(x)
 +
\right) y = 0,
 +
\end{equation}
 
in the form
 
in the form
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\begin{equation}\label{eq2}
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\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
 +
\[
 +
\eta(\xi) = \Phi(x)y(x),\quad
 +
\xi = \int_\alpha^x \sqrt{r(t)}\,\mathrm{d}t, \quad
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\Phi(x) = r(x)^{1/4}
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\exp\left(
 +
\frac{1}{2}\int_\alpha^x p(t)\,\mathrm{d}t
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\right),
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\]
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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}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596402.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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====References====  
 
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{|
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596403.png" /> is parameter. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596406.png" />, then equation (1) reduces to the Liouville normal form (2) by means of the substitution
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|-
 
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|valign="top"|{{Ref|In}}||valign="top"| E.L. Ince, "Ordinary differential equations", Dover, reprint (1956)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596407.png" /></td> </tr></table>
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|-
 
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|valign="top"|{{Ref|Ka}}||valign="top"| E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", '''1. Gewöhnliche Differentialgleichungen''', Chelsea, reprint (1947)
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596408.png" /></td> </tr></table>
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|-
 
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|valign="top"|{{Ref|Li}}||valign="top"| J. Liouville, ''J. Math. Pures Appl.'', '''2''' (1837) pp. 16–35
which is called the Liouville transformation (introduced in [[#References|[1]]]). The Liouville normal form plays an important role in the investigation of the asymptotic behaviour of solutions of (1) for large values of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059640/l0596409.png" /> or the argument, and in the investigation of the asymptotics of eigen functions and eigen values of the [[Sturm–Liouville problem|Sturm–Liouville problem]] (see [[#References|[3]]]).
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|-
 
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|valign="top"|{{Ref|Ti}}||valign="top"| E.C. Titchmarsh, "Eigenfunction expansions associated with second-order differential equations", '''1–2''', Clarendon Press (1946–1948)
====References====
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|-
<table><TR><TD valign="top">[1]</TD> <TD valign="top"J. Liouville,   ''J. Math. Pures Appl.'' , '''2''' (1837) pp. 16–35</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.C. Titchmarsh,   "Eigenfunction expansions associated with second-order differential equations" , '''1–2''' , Clarendon Press (1946–1948)</TD></TR></table>
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|}
 
 
 
 
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,  "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>
 

Latest revision as of 02:23, 30 July 2012

2010 Mathematics Subject Classification: Primary: 34A30 Secondary: 34B24 [MSN][ZBL]

$ \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 [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]).

References

[In] E.L. Ince, "Ordinary differential equations", Dover, reprint (1956)
[Ka] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden", 1. Gewöhnliche Differentialgleichungen, Chelsea, reprint (1947)
[Li] J. Liouville, J. Math. Pures Appl., 2 (1837) pp. 16–35
[Ti] E.C. Titchmarsh, "Eigenfunction expansions associated with second-order differential equations", 1–2, Clarendon Press (1946–1948)
How to Cite This Entry:
Liouville normal form. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Liouville_normal_form&oldid=27240
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article