We show how Laplace Transforms may be used to solve initial value problems with piecewise continuous forcing functions.

### Constant Coefficient Equations with Piecewise Continuous Forcing Functions

We’ll now consider initial value problems of the form

where , , and are constants () and is piecewise continuous on . Problems of this kind occur in situations where the input to a physical system undergoes instantaneous changes, as when a switch is turned on or off or the forces acting on the system change abruptly.It can be shown that the differential equation in (eq:8.5.1) has no solutions on an open interval that contains a jump discontinuity of . Therefore we must define what we mean by a solution of (eq:8.5.1) on in the case where has jump discontinuities. The next theorem motivates our definition. We omit the proof.

- (a)
- and .
- (b)
- and are continuous on .
- (c)
- is defined on every open subinterval of that does not contain any of the points …, , and on every such subinterval.
- (d)
- has limits from the right and left at … .

We define the function of Theorem thmtype:8.5.1 to be the solution of the initial value problem (eq:8.5.1).

We begin by considering initial value problems of the form

where the forcing function has a single jump discontinuity at .We can solve (eq:8.5.2) by the these steps:

*Step 1.* Find the solution of the initial value problem
*Step 2.* Compute and . *Step 3.* Find the solution of the initial value problem
*Step 4.* Obtain the solution of (eq:8.5.2) as

It can be shown that exists and is continuous at . The next example illustrates this procedure.

If and are defined on , we can rewrite (eq:8.5.2) as and apply the method of Laplace transforms. We’ll now solve the problem considered in Example example:8.5.1 by this method.

### Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)