 Welcome back to Mechanical EI. Did you know that many applications of boundary value problems, as in the case of electromagnetic potential, can be solved using Laplace transforms? This makes us wonder, how can applications of boundary value problems be solved using Laplace transforms? Before we jump in, check the previous part of the series to learn about what convolution theorem is. Now, a boundary value problem is a differential equation together with a set of additional constraints called the boundary conditions. To solve these equations, the procedure is pretty simple and can be done using the following method. Let capital Y of s equal Laplace transform of Y of t of s. Instead of solving directly for Y of t, we derive a new equation for capital Y of s and once we find capital Y of s, we can inverse transform it to determine Y of t. It is best illustrated with an example. Consider that we want to solve the differential equation f double dash of t plus 4 into f of t equals sin of 2t, with initial conditions f of 0 equals 0 and f dash of 0 equals 0. We know that if psi of t equals sin of 2t, then Laplace transform of psi of t equals 2 upon s squared plus 4. Applying Laplace transform to both sides of the equation, we get the following equation. Taking Laplace transform of f of t common in rearranging, we get Laplace transform of f of t equals 2 upon s squared plus 4, the whole square. Finally, we apply Laplace inverse transform to get the value of f of t and solve the equation. Hence, we first saw how Laplace transforms can be used to solve boundary value problems and then went on to see an example of it. Like, subscribe and comment with your feedback to help us make better videos. Thanks for watching. Also, thanks a lot for those constructive comments. You helped the channel grow. So here are the top mechanical EIs of our last videos. In the next episode of Mechanical EI, find out what complex variables are.