Our goal is to apply the Laplace transform to both sides of a differential equation. In order to do this, we need to understand \(\mathscr{L}\left\{f^{(n)}(t)\right\}\text{.}\)
If \(f\) is continuous and piecewise smooth for \(t \geq 0\) and of exponential order as \(t \rightarrow \infty\) with constant \(c\text{,}\) then \(\mathscr{L}\{f'(t)\}\) exists for \(s \gt c\) and is given by
Use linearity, the table of Laplace transforms, and our knowledge of the intial values to write an equation in the function \(X(s)\text{.}\) (As we saw in the previous section, you need the initial values to compute the Laplace transform of any derivatives in the differential equation.)
Use \(\mathscr{L}^{-1}\) to find \(x(t)\text{.}\) At this step, you will often need to use partial fractions so that you can use the linearity of \(\mathscr{L}^{-1}\) and the table of Laplace transforms to find \(x(t)\text{.}\)NOTE: Example 2 in Section 7.2 of your textbook by Edwards, et.al. may be helpful for speeding up your some partial fractions computations.
The formula in this section is not on the table of Laplace transforms. In the past, I have been told that it will be given to you on the final exam if you need it. The next theorem is Theorem 2 in Section 7.2 of your textbook by Edwards, et.al.
If \(f\) is continuous and piecewise smooth for \(t \geq 0\) and of exponential order as \(t \rightarrow \infty\) with constant \(c\text{,}\) then for \(s \gt c\)
