In this section we consider \(\mathscr{L}\{e^{at}f(t)\}\text{.}\) The next theorem, which is Theorem 1 in Section 7.3 of your textbook by Edwards, et. al., shows us that \(\mathscr{L}\{e^{at}f(t)\}\) is a horizontal shift of \(F(s)\) on the \(s\)-axis.
If \(F(s)=\mathscr{L}\{f(t)\}\) exists for some \(s \gt c\text{,}\) then \(\mathscr{L}\{e^{at}f(t)\}\) exists for \(s \gt a+c\text{.}\) Moreover, for \(s \gt a+c\text{:}\)
In the examples in this section, we will need to apply the technique of partial fractions to \(F(s)\) so that we can evaluate \(\mathscr{L}^{-1}\{F(s)\}\text{.}\) You learned about partial fractions in Calculus II. Perhaps the approach that we will take in these examples will help you to become more efficient at finding the constants in a partial fractions decomposition. If you have learned good techniques for finding the constants in other classes, you are more than welcome to use them.
Example226.Partial fractions with repeated linear factors.
This example shows you a tool that can be employed when your denominator has repeated linear factors. Evaluate \(\mathscr{L}^{-1}\{\frac{1}{(s^2+s-6)^2}\}\text{.}\)
