Solving Systems of Linear Differential Equations

Abstract

Michael Olinick received his Ph.D. in mathematics from the University of Wisconsin, and is currently Assistant Professor of Mathematics at Middlebury College, Middlebury, Vermont. His main research interest is topology, and he has several publications in that area. He has also been experimenting in designing undergraduate courses on mathematical models in the social sciences. A common theme in revisions of the undergraduate mathematics curriculum is the earlier introduction of linear algebra. One approach to this is a course, to follow a traditional twoor three-semester calculus sequence, that develops sufficient linear algebra to treat systems of first-order linear differential equations. Such a treatment of linear algebra and differential equations may include (1) the basic matrix operations, including elementary row operations; (2) solution of linear systems of algebraic equations by reducing the coefficient matrix to row echelon form; (3) vector space concepts motivated by examining the solution sets to the algebraic systems; (4) elementary properties of the determinant; (5) linear systems of differential equations with emphasis on the vector space structure of the solution set; and (6) methods of solution of linear systems of differential equations with constant coefficients. Two texts [1, 2] have been written explicitly for such a course and others could be adapted to it [3, 4]. To introduce this quantity of material in a one-quarter or one-term course, some of the topics can not be treated with complete mathematical rigor. A typical choice is to omit a justification for the procedure to solve a system of differential equations with constant coefficients when the coefficient matrix is not diagonable. This note will discuss how this topic can be presented without elaborate algebraic machinery. We wish to consider systems of linear differential equations