An implementation of Kovacic's algorithm for solving second order linear homogeneous differential equations

ABSTRACT

Kovacic [3] has given an algorithm for the closed form solution of differential equations of the form ay" + by' + cy &equil; 0, where a, b, and c are rational functions with complex coefficients of the independent variable x. The algorithm provides a Liouvillian solution (i.e. one that can be expressed in terms of integrals, exponentials and algebraic functions) or reports that no such solution exists. In this note a version of Kovacic's algorithm is described. This version has been implemented in MACSYMA and tested successfully on examples in Boyce and DiPrima [1], Kamke [2], and Kovacic [3]. Modifications to the algorithm have been made to minimize the amount of code needed and to avoid the complete factorization of a polynomial called for. In Section 2 these issues are discussed and in Section 3 the author's current version of the algorithm is described.

REFERENCES

• 1: Boyce, William E. and DiPrima, Richard C., Elementary Differential Equations and Boundary Value Problems, New York, J. Wiley, (1965).
• 2: Kamke, E., Differentialgleichungen Losungsmethoden und Losungen, New York, Chelsea Publishing Co., (1948).
• 3: Kovacic, Jerald J., "An Algorithm for Solving Second Order Linear Homogeneous Differential Equations", to appear.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Nonalgebraic algorithms

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.3 Numerical Linear Algebra
      G.1.7 Ordinary Differential Equations
          Subjects: Boundary value problems

General Terms:
Algorithms, Theory