Liouville normal form

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A way of writting a second-order ordinary linear differential equation


in the form


where is parameter. If , and , then equation (1) reduces to the Liouville normal form (2) by means of the substitution

which is called the Liouville transformation (introduced in [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 or the argument, and in the investigation of the asymptotics of eigen functions and eigen values of the Sturm–Liouville problem (see [3]).


[1] J. Liouville, J. Math. Pures Appl. , 2 (1837) pp. 16–35
[2] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1947)
[3] E.C. Titchmarsh, "Eigenfunction expansions associated with second-order differential equations" , 1–2 , Clarendon Press (1946–1948)



[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
How to Cite This Entry:
Liouville normal form. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article