Laplace transform applied to differential equations
Encyclopedia
The use of Laplace transform makes it much easier to solve linear differential equations
with given initial conditions.
First consider the following relations:
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
with given initial conditions.
First consider the following relations:



Consider the following differential equation:
This equation is equivalent to
which is equivalent to
Note that the are initial conditions.
The solution for f(t) will be given by applying the Laplace inverse transform to
An example
We want to solve
with initial conditions f(0) = 0 and f ′(0)=0.
We note that
and we get
So this is equivalent to
We deduce
So we apply the Laplace inverse transform and get

