The Laplace transform is what is called an integral transform, and it trans-

forms a function f(t), in which t is often the time, into a related function

F (s), in which s is the transform variable. When applied to ordinary linear

differential equations, the Laplace transform provides an efficient method for the solution of initial value problems. The Laplace transform approach is

unlike the method just described in which both a complementary function

and a particular integral need to be determined in order to form a general

solution that must then be matched to the initial conditions. When using the

Laplace transform to solve linear differential equations, the initial conditions

are incorporated at the time the transformation is carried out. As a result, the

answer obtained is the solution to the given initial value problem, and so it is

free from arbitrary constants. Before defining the Laplace transform, we first need to introduce a new

term. A function f(t) is said to be piecewise continuous on the interval

a0/t0/b if the interval can be subdivided into smaller intervals on each of which f (t ) is continuous, but across each point separating adjacent sub-

intervals f(t) experiences a finite jump. A typical piecewise continuous func-

tion f (t ) is shown in Fig. 158, in which f(t) is continuous for a0/t0/b except at t1 and t2 where finite jumps occur.