Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition

The Laplace Transform

n integral transform is a relation of the form

which transforms or maps a given function f (t) into another function F(s). The function K(t, s) in Eq. (1) is called the kernel of the transform and the function F(s) is called the transform of f(t). It is possible that α = −∞ or β = ∞, or both. If α = 0, β = ∞, and K(t, s) = e^−st, Eq. (1) takes the form

In this case, F(s) is called the Laplace transform of f(t).

In general, the parameter s can be a complex number but in most of this chapter we assume that s is real. It is customary to refer to f(t) as a function, or signal, in the time or “t-domain” and F(s) as its representation in the “s-domain.” The Laplace transform is commonly used in engineering to study input–output relations of linear systems, to analyze feedback control systems, and to study electric circuits. One of its primary applications is to convert the problem of solving a constant coefficient linear differential equation in the t-domain into a problem involving algebraic operations in the s-domain. The general idea, illustrated by the block diagram ...