# Solving linear two variable functional equations with computer

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## Summary.

In the present paper we deal with the linear two variable functional equation
\( h_{0}(x,y)f_{0}(g_{0}(x,y)) + \cdots + h_{n}(x,y)f_{n}(g_{n}(x,y)) = F(x,y) \)
where *n* is a positive integer,
\( g_{0}, g_{1}, \ldots, g_{n}, h_{0}, h_{1}, \ldots, h_{n}, \)
and *F* are given real valued analytic functions on an open set
\( \Omega \subset \mathbb{R}^2, \)
furthermore
\( f_{0}, f_{1}, \ldots, f_{n} \)
are unknown functions.

Applying the results of Páles [3] we get recursively an
inhomogeneous linear differential-functional equation in one of
unknown function for *f*_{0}, then for
\( f_{1}, f_{2}, \ldots, f_{n}, \)
respectively.

One of our main result states that the solutions of the differential-functional equation obtained are the same as that of an ordinary differential equation (under some assumptions), whose order is usually much smaller than the order of the differential-functional equation.

Our aim is also to describe a computer-program which solves functional equations of this type.

The theoretical background of the program is based on the result and algorithm of Páles [3] and our main results. This algorithm is implemented in Maple VR4 symbolic language.

## Mathematics Subject Classification (2000).

39B22.## Keywords.

Functional equation.## Preview

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