A note on the Laplace transform method for initial value problems

Abstract

This note provides a tutorial treatment of the basic (unilateral) Laplace transform method for solving initial value problems for linear constant coefficient differential equations with terms that contain derivatives of the input. The input is allowed to contain jump discontinuities (steps) and so it would first appear that one needs the advanced machinery of generalized functions (distributions) to apply the Laplace transform method here. This is, however, not the case. It suffices to use the basic Laplace transform for ordinary functions, where the integral is interpreted as the most basic integral notion in calculus, i.e. as a Riemann integral. There is thus no need to specify any convention on the lower limit of integration as they will all give the same transform. Furthermore, it suffices to use the derivative rule in the basic form L[f′](s)=sL[f](s)−f(0) for smooth enough differentiable functions. Finally, we point out important limitations of the bilateral Laplace transform method in input–output stability analysis of feedback systems.