Given , the Laplace transform of some function , we study techniques for recovering the function .

The Inverse Laplace Transform

Definition of the Inverse Laplace Transform

In Trench 8.1 we defined the Laplace transform of by We’ll also say that is an inverse Laplace Transform of , and write To solve differential equations with the Laplace transform, we must be able to obtain from its transform . There’s a formula for doing this, but we can’t use it because it requires the theory of functions of a complex variable. Fortunately, we can use the table of Laplace transforms to find inverse transforms that we’ll need.

The next theorem enables us to find inverse transforms of linear combinations of transforms in the table. We omit the proof.

Inverse Laplace Transforms of Rational Functions

Using the Laplace transform to solve differential equations often requires finding the inverse transform of a rational function where and are polynomials in with no common factors. Since it can be shown that if is a Laplace transform, we need only consider the case where . To obtain , we find the partial fraction expansion of , obtain inverse transforms of the individual terms in the expansion from the table of Laplace transforms, and use the linearity property of the inverse transform. The next two examples illustrate this.

The shortcut employed in the second solution of Example example:8.2.4 is Heaviside’s method. The next theorem states this method formally.

Some software packages that do symbolic algebra can find partial fraction expansions very easily. We recommend that you use such a package if one is available to you, but only after you’ve done enough partial fraction expansions on your own to master the technique.

Text Source

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