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 formula_8 and substituting formula_9 with formula_10 one obtains
The solution for "f"("t") is obtained by applying the inverse Laplace transform to formula_12
Note that if the initial conditions are all zero, i.e.
then the formula simplifies to