Using Laplace Transforms

Lesson 7F

Here is the process for solving an initial value problem using Laplace transforms:

  1. Take the Laplace transform of both sides.
  2. Simplify so things are in terms of \(\mathcal L\{y\}\) only. Remember, we will often write \(Y\) instead of \(\mathcal L\{y\}\).
  3. Solve for \(Y\).
  4. Take the inverse Laplace transform of both sides.

Usually, you will use a chart to take inverse Laplace transforms. For the problems below, the only Laplace transform you need to know is \[\mathcal L\{e^{at}\} = \frac{1}{s-a}. \]

Practice

Try solving this initial value problem: \[ y'' + y' - 12y = e^{2t}, \quad y(0)=0, \quad y'(0)=0 \]

Review these, you’ll need them soon.

A lot of the difficulty in dealing with Laplace transforms is taking the inverse Laplace transform. To do this, you need to make your function look exactly like something from the chart. Here are two algebra tools that you will use to do this:

  1. Partial fractions decomposition. If you don’t remember how to do them, look at these notes on the coverup method.

  2. Completing the square. If you don’t remember how to do this, look at these notes.