Defining equation

determining equation, characteristic equation

An equation associated with a regular singular point $z = a$ of an ordinary linear differential equation

$$\tag{1 } p _ {0} ( z) w ^ {(} n) + \dots + p _ {n} ( z) w = 0.$$

Let

$$p _ {j} ( z) = ( z- a) ^ {n-} j q _ {j} ( z),$$

where the functions $q _ {j} ( z)$ are holomorphic at the point $z= a$ and $q _ {0} ( a) \neq 0$. The defining equation takes the form:

$$\tag{2 } \lambda \dots ( \lambda - n+ 1) q _ {0} ( a) + \dots + \lambda q _ {n-} 1 ( a) + q _ {n} ( a) = 0.$$

If the roots $\lambda _ {j}$, $1 \leq j \leq n$, of equation (2) are such that all differences $\lambda _ {j} - \lambda _ {k}$, where $j \neq k$, are not integers, then equation (1) has a fundamental system of solutions of the form

$$\tag{3 } w _ {j} ( z) = ( z- a) ^ {\lambda _ {j} } \phi _ {j} ( z),\ 1 \leq j \leq n ,$$

where the functions $\phi _ {j} ( z)$ are holomorphic at $z= a$. Otherwise the coefficients $\phi _ {j} ( z)$ can be polynomials in $\mathop{\rm ln} ( z- a)$ with coefficients holomorphic at $z= a$.

The defining equation for a system of $n$ equations

$$\tag{4 } ( z- a) w ^ \prime = A( z) w,$$

corresponding to the regular singular point $z= a$, takes the form

$$\mathop{\rm det} \| \lambda I - A( a) \| = 0,$$

where $A( z)$ is a matrix-function of order $n \times n$, holomorphic at $z= a$ and $A( a) \neq 0$. If all differences $\lambda _ {j} - \lambda _ {k}$, where $j \neq k$, are not integers, where the $\lambda _ {j}$ are the eigen values of $A$, then the system (4) has a fundamental system of solutions of the form (3), where $\phi _ {j} ( z)$ are vector-functions holomorphic at $z= a$; otherwise, the vector-functions $\phi _ {j} ( z)$ can be polynomials in $\mathop{\rm ln} ( z- a)$ with coefficients which are vector-functions holomorphic at $z= a$.

In another sense, the term "determining equation" is used in research on transformation groups admitted by ordinary partial differential equations (see [3]).

References

[1]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955)
[2]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
[3]	L.V. [L.V. Ovsyannikov] Ovsiannikov, "Group analysis of differential equations" , Acad. Press (1982) (Translated from Russian)

Comments

A defining equation is more commonly called an indicial equation.