Background. Laplace transforms are useful in solving initial value problems for ordinary differential equations and systems of ordinary differential equations. In this exercise we explore the method of using Laplace transforms. First load Mathematica's built in "LaplaceTransform" subroutine package.

Computer Lab Work.

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

and plot the solution over the interval 0 <= t <= 7.

First set up the initial conditions and function 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.
We could look them up in a table or let the following loop do it for us.

Find the inverse transform and plot the solution.

Exercise 2. Use Laplace transforms to solve the initial value problem

and plot the solution over the interval 0 <= t <= 7.

First set up the initial conditions and function f(t).

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

Find the inverse transform and plot the solution.

Exercise 3. Use Laplace transforms to solve the initial value problem for the system of D.E.'s

and plot the solution for 0 <= t <= 5.5

First set up the initial conditions and functions f(t) and g(t)

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

Find the inverse transforms and plot the solution.

Solutions.

Return to the Differential Equations Project

Return to the Numerical Analysis Project

Return to the Complex Analysis Project

(c) John H. Mathews, 1998