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# Laplace transform applied to differential equations

The use of Laplace transform makes it much easier to solve linear differential equations.

First consider the folowing relations :

Suppose we want to solve the given differential equation:

this equation is equivalent to :

which is equivalent to :

note that the are initial conditions.

Then all we need to find f(t) is to apply the Laplace inverse transform to

## An example

We want to solve :

with initial conditions f(0) = 0 and f ′(0)=0

we note :

and we get :

so this is equivalent to :

we deduce :

So we apply the Laplace inverse transform and get

$f\left(t\right)=\\frac\left\{1\right\}\left\{8\right\}\\sin\left(2t\right)-\\frac\left\{t\right\}\left\{4\right\}\\cos\left(2t\right)$