Application to Differential Equations

Consider the linear differential equation with constant coefficients

displaymath99

under the initial conditions

displaymath100

The Laplace transform directly gives the solution without going through the general solution. The steps to follow are:

(1)
Evaluate the Laplace transform of the two sides of the equation (C);
(2)
Use Property 14 (see Table of Laplace Transforms)

displaymath101;

(3)
After algebraic manipulation, write down

displaymath102;

(4)
Make use of the properties of the inverse Laplace transform tex2html_wrap_inline109, to find the solution y(t).

Example: Find the solution of the IVP

displaymath113,

where

displaymath115.

Solution: Let us follow these steps:

(1)
We have

displaymath117;

(2)
Using properties of Laplace transform, we get

displaymath119,

where tex2html_wrap_inline121 . Since tex2html_wrap_inline123 , we get

displaymath125;

(3)
Inverse Laplace:

Using partial decomposition technique we get

displaymath127,

which implies (see Table of Laplace Transforms)

displaymath129

Since

displaymath131,

which gives (see Table of Laplace Transforms)

displaymath133,

and

displaymath135

Hence,

displaymath137

If you would like more practice, click on Example.

[Differential Equations] [First Order D.E.] [Second Order D.E.]
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Author: Mohamed Amine Khamsi

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