# Polynomial Solutions of Differential Equations

Polynomial Solutions of Differential Equations. Masters thesis, King Fahd University of Petroleum and Minerals.

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Polynomial solutions of linear differential equations have been investigated by many authors. In this thesis, we study a new approach based on elementary linear algebra for investigating polynomial solutions of differential equations $L(y)= \sum\limits_{k=0}^N a_{k}(x)y^{(k)}=0$ with polynomial coefficients. Any differential operator of the form $L(y)= \sum\limits_{k=0}^N a_{k}(x)y^{(k)}$, where $a_{k}$ is a polynomial of degree $\leq k$, on the space of polynomials over an infinite field $F$, has all eigenvalues in $F$. If these eigenvalues are distinct, then there is a unique monic polynomial of degree $n$ which is an eigenfunction of the operator $L$. We also carry out a study of orthogonality of such eigenfunctions. Further for the general case of operators $L(y)= \sum\limits_{k=0}^N a_{k}(x)y^{(k)}$ where degree of the polynomials $a_{k}$ is arbitrary, an algorithmic procedure is presented for determining the existence of polynomial solutions as well as for constructing these solutions. An implementation of the algorithmic procedure is carried out through Maple codes which are applied to obtain polynomial solutions of different types of differential equations of current interest in Physics.