Laplace transform applied to differential equations

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.

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 exampleEdit

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

 

BibliographyEdit

  • 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