To solve a differential equation using Laplace Transforms with Mathematica, we first must load in the package Calculus`LaplaceTransform` as shown below. The basic procedure is to apply the transform to both sides of the differential equation, yielding an equation in terms of Y( s ). Next, solve for Y( s ) using the Mathematica function, Solve. In certain cases, the NSolve, numeric solve, would be more appropriate. The final step is to apply the inverse transform to Y( s ), giving the solution y( t ). This is illustrated in the Mathematica notebook below.

The first two examples illustrate the LaplaceTransform and InverseLaplaceTransform syntax. What follows is a spring motion problem with forced motion in the form of a unit step function. Here the external driving force of 2 is applied for 10 seconds and then turned off. Note what happens to the spring as soon as the external force is removed.

Also note that the spring remains at 1/3 as long as the external force is applied. The 1/3 comes from the external force of 2 divided by the spring constant of 6. This effect was discovered by Geoff St. Pierre.

1. Verify the St. Pierre effect by changing k to 1 and plotting the solution.

2. Change the external force to sin( t ) applied for 5 seconds and then shut off.

3. Change the external force to sin( t ) applied forever after 5 seconds of no external force.

4. Experiment with external forces that involve the unit step function.