Approach
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
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

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