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.
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
:
An example
We want to solve
:
with initial conditions f(0) = 0 and f′(0)=0.
We note that
:
and we get
:
The equation is then equivalent to
:
We deduce
:
Now we apply the Laplace inverse transform to get
:
Bibliography
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. {{isbn|1-58488-299-9}}
Category:Differential equations
Category:Differential calculus