Application to Differential Equations
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:
- Evaluate the Laplace transform of the two sides of the
- Use Property 14 (see Table of Laplace Transforms)
- After algebraic manipulation, write down
- Make use of the properties of the inverse Laplace transform
, to find the solution y(t).
Example: Find the solution of the IVP
Solution: Let us follow these steps:
- We have
- Using properties of Laplace transform, we get
where . Since , we get
- Inverse Laplace:
partial decomposition technique we
which implies (see Table of Laplace Transforms)
which gives (see Table of Laplace Transforms)
If you would like more practice, click on Example.
[First Order D.E.]
[Second Order D.E.]
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