Polynomial solutions to second order linear homogeneous ordinary differential equations. Properties and approximation. (English) Zbl 0787.34018
The author deals with nontrivial polynomial solutions of a second order homogeneous linear differential equation with polynomial coefficients. He first discusses the existence of a solution and its uniqueness (up to a multiplicative constant), and then shows a method of approximating all zeros of the solution polynomial. The method consists in solving a certain system of nonlinear equations derived directly from the coefficients of the original differential equation. Newton’s iteration is recommended to solve the system. The author points out the possibility of approaching in this way all zeros of known orthogonal polynomials, such as Hermite or Laguerre polynomials. For the numerical experiments in this case, the reader is referred to a forthcoming paper of the author.
MSC:
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 34A30 | Linear ordinary differential equations and systems |
| 33C65 | Appell, Horn and Lauricella functions |
| 41A10 | Approximation by polynomials |
| 65L99 | Numerical methods for ordinary differential equations |
Keywords:
polynomial solutions; second order homogeneous linear differential equation with polynomial coefficients; existence; uniqueness; zeros; Newton’s iteration; Hermite or Laguerre polynomialsReferences:
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