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