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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
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
while in an electrical system, the parameters are
The function is the forcing function and the solution is
called the response of the system.
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 .