Module for the Laplace Transform Solution of Initial Value Problems

Numerical Methods for O. D. E.  using Mathematica. (c) John H. Mathews, 2003

Background.

    Laplace transforms are useful in solving initial value problems for ordinary differential equations and systems of ordinary differential equations. Step functions are used in various applications involving electrical circuits and mechanical vibrations.  In this exercise we explore how to make pulses from step functions and the use the method of using Laplace transforms to find the solution to the D. E.  First load Mathematica's built in "LaplaceTransform" subroutine package.

The Unit Step Function. Let .  Then, the unit step function is

              

Example 1.  Use Laplace transforms to solve the initial value problem  

      with  ,

where  f(t)  is consists of  6  pulses given by

    .  

Example 2.  Solve the initial value problem  

      with  ,

where  f(t)  is consists of  2  pulses given by  

    .