Laplace transform applied to differential equations

In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain.

The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions.

First consider the following property of the Laplace transform: One can prove by induction that Now we consider the following differential equation: with given initial conditions Using the linearity of the Laplace transform it is equivalent to rewrite the equation as obtaining Solving the equation for

Note that if the initial conditions are all zero, i.e. then the formula simplifies to We want to solve

We note that and we get The equation is then equivalent to We deduce Now we apply the Laplace inverse transform to get