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.

Computer Lab Work.

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

where f[t] is consists of 6 pulses given by the following construction:

First set up the initial conditions and find the Laplace transform of f(t).

Second, set up the D.E. in Laplace transformation format and solve for Y[s].

Since Y[s] is in its expanded form, we can find the inverse transform of each term in the above sum is done in the following loop. In order to rapidly find the inverse Laplace transform of Y[s] it speeds things up to find the inverse of each term in the sum and add them up as we go along. The answer is stored in the variable named "sum".

Now plot the solution over the interval 0 <= t <= 21.

Exercise 2. Solve the initial value problem

where f[t] is consists of 2 pulses given by the following construction:

First set up the initial conditions and find the Laplace transform of f(t).

Second, set up the D.E. in Laplace transformation format and solve for Y[s].

Since Y[s] is in its expanded form, we can find the inverse transform of each term in the above sum is done in the following loop. In order to rapidly find the inverse Laplace transform of Y[s] it speeds things up to find the inverse of each term in the sum and add them up as we go along. The answer is stored in the variable named "sum".

Now plot the solution over the interval 0 <= t <= 12.

Solutions.

Return to the Differential Equations Project

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(c) John H. Mathews, 1998