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
Solving the equation for and substituting with one obtains
The solution for f(t) is obtained by applying the inverse Laplace transform to
Note that if the initial conditions are all zero, i.e.
then the formula simplifies to
We want to solve
with initial conditions f(0) = 0 and f′(0)=0.
We note that
and we get
The equation is then equivalent to
Now we apply the Laplace inverse transform to get