Consider the linear differential equation with constant coefficients

under the initial conditions

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)*;

**(3)***After algebraic manipulation, write down*;

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

**Example:** Find the solution of the IVP

,

where

.

**Solution:** Let us follow these steps:

**(1)**- We have
;

**(2)**- Using properties of Laplace transform, we get
,

where . Since , we get

;

**(3)**- Inverse Laplace:
Using partial decomposition technique we get

,

which implies (see Table of Laplace Transforms)

Since

,

which gives (see Table of Laplace Transforms)

,

and

Hence,

**If you would like more practice, click on Example.**

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