# The Initial Value Problem of Ordinary Differential Equations with Constant Coefficients

• Gustav Doetsch

## Abstract

An important application of the Differentiation Theorem 9.1 and the Convolution Theorem 10.1accrues when these are called to aid with the problem of integrating ordinary linear differential equations with constant coefficients in the interval t ≧ 0, for specified values of the solution and some of its derivatives at t = 0, the initial values (Initial Value Problem). This is a problem which may be solved by a familiar classical technique: First one produces a sufficient number of fundamental solutions of the homogeneous equation, then one constructs the general solution as a linear combination of these; by means of the “variation of the constants” one seeks a special solution of the inhomogeneous equation. Addition of the latter to the former yields the general solution of the inhomogeneous equation. Lastly one must “adjust” the arbitrary coefficients so that the general solution agrees with the specified initial values. Theoretically, this technique seems simple; however it creates great difficulties in practice, particularly when applied to differential equations of higher order. By contrast, we shall observe that the method based upon the ℒ-transformation provides the solution of such problems with a minimum of technical effort.

### Keywords

Convolution Verse