The Laplace transform allows us to solve initial value problems arising from mass/spring/dashpot systems with a piecewise driving force. For example, consider the following IVP that models a mass/spring system (no resistance):

where:

Applying the Laplace transform to the
IVP and solving for *Y*(*s*), the Laplace transform of
*y*, we obtain:

Hence, applying the inverse Laplace
transform to *Y*, we find the solution to the IVP is:

The pulse driving force *f* (*t*) and solution
*y* (*t*) are shown below.

We also present another example. This one includes resonance. Consider the following IVP that also models a mass/spring system (no resistance):

where:

As before, we apply the Laplace
transform to the IVP and solve for *Y*(*s*), the Laplace
transform of *y*. We obtain:

Hence, applying the inverse Laplace
transform to *Y*, we find the solution to the IVP is:

The pulse driving force *f* (*t*) and solution
*y* (*t*) are shown below.

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