Applications of boundary value problems | Laplace Transforms

Welcome back MechanicaLEi, Did you know that many applications of boundary value problem 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 out the previous part of this 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 by 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 zero equals zero and f dash of zero equals zero. We know that if psi of t equals sine 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 of f of t common and rearranging, we get Laplace transform of f of t equals two upon s squared plus 4 the whole squared. 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 to it?

In the next episode of MechanicaLEi find out what complex variables are?

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