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# Applying the Laplace Transform to Differential Equations

The Laplace transform may be used to simplify the solving of linear differential equations with constant
coefficients. Starting with a very simple example, suppose we have
the differential equation

.

To solve this using the Laplace transform, assume we take the Laplace transform of both sides:

or, by linearity

and by using the rule for the Laplace transform of derivatives:

.

By using the initial conditions, we get:

.

By solving this equation for , we get that

.

The question is, what function has as
its Laplace transform? This process is called finding the *inverse Laplace transform* of .

In this example, we can use partial fraction decomposition to aid us: by observing that , and
therefore . Using the *cover-up* method,
one may quickly determine that .

To do this in Maple, use the command:

> convert( (s + 3)/(s^2 + 3*s + 2), parfrac, s );

However, by inspection, we observe that
and
; therefore
.

Consequently, .

# Remark 1

The above example is a specific example of the more general 2nd-order differential equation

where all of the parameters are given parameters. In a mechanical spring system, the parameters
are
,
, and
while in an electrical system, the parameters are
,
, and
.

The function is the forcing function and the solution is
called the *response* of the system.

# Remark 2

The same situation applies to systems of 1st-order linear differential equations with constant
coefficients. See AMEM, p.126-8 to see the solution to the system

where and .