Solving Differential Equations with Laplace Transforms

In this blog post, you will learn what a Laplace Transform is, and how to use them to solve differential equations.

What is a Laplace Transform?

Laplace Transforms are important because they enable you to solve differential equations algebraically. Therefore, they are important for solving problems in science, engineering, and applied math.

The definition of a Laplace Transform is the following:

Given a function f(t) defined for all t greater than or equal to zero, the Laplace Transform of f is the function F defined as follows:

For all values of s for which the improper integral converges.

Computing the Laplace Transform by Definition

Let’s look at a simple function f(t)=2, and compute its Laplace Transform.

Note: once you compute some of these by hand, you notice you keep using the same ones over and over. Therefore, there are tables of Laplace transforms you can reference to compute them. You can also use them for computing the inverse of the transform.

Inverse Transforms

As you might guess, sometimes, we want to reverse F(s) back to a function f(t), which is the inverse Laplace Transform. Since, we just computed one, let’s reverse it.

This is important when you want to transform the algebraically solved expression back into a particular solution of a differential equation (more on that later.)

Solving a Differential Equation with Laplace Transforms

Consider the following differential equation:

with the initial conditions x(0)=x'(0)=0. Taking the Laplace transform of it converts it into the following:

Solving for X(s), we get the following:

Now, to compute the inverse Laplace transform, which will give us x(t). We must use partial fractions, which gives us the following:

Solving this partial fraction, you will obtain A=C=0, B= 3/5, and D=-3/5.

Therefore, we have the following:

By computing the inverse Laplace transform, we get x(t), which is the following:

This is the particular solution to the equation given the initial conditions.

Final Thoughts

In this blog post, you learned what Laplace transforms are, and you learned how to use them to solve differential equations.