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One of the advantages of using the Laplace Transform to solve differential equations is that all initial conditions are automatically included during the process of transformation, so one does not have to find the homogeneous solutions and the particular solution separately.

When solving initial-value problems using the Laplace transform, we perform the following steps in sequence:

1) Apply the Laplace Transform to both sides of the equation.
2) Solve the resulting algebra problem from step 1.
3) Apply the Inverse Laplace Transform to the solution of 2.

Example

Consider a mass-spring system with a forcing function :

Step One: Apply the Laplace Transform to both sides of the equation.

Linearity

Step Two: Solve the algebra problem.

Step Three: Apply the Inverse Laplace Transform to obtain the final solution.

Linearity
Convolution

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