University of Florida/Egm4313/s12.team11.imponenti/R3.1

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Report 3, Problem 1[edit]

Problem Statement[edit]

Find the solution to the following L2-ODE-CC:

With the following excitation:

And the following initial conditions:

Plot this solution and the solution in the example on p.7-3

Homogeneous Solution[edit]

To find the homogeneous solution we need to find the roots of our equation

We know the homogeneous solution for the case of a real double root with to be

Particular Solution[edit]

For the given excitation we must use the Sum Rule to the particular solution as follows

where and are the solutions to and , respectively

First Particular Solution[edit]

,

from table 2.1, K 2011, pg. 82 we have

but this corresponds to one of our homogeneous solutions so we must use the modification rule to get

Plugging this into the original L2-ODE-CC then substituting;

so and the first particular solution is,

Second Particular Solution[edit]

,

from table 2.1, K 2011, pg. 82 we have

Plugging this into the original L2-ODE-CC then substituting;

grouping like terms we get three equations to solve for the three unknowns, these are written in matrix form



solving by back subsitution leads to

so the second particular solution is,

General Solution[edit]

The general solution is the summation of the homogeneous and particular solutions

Applying the first initial condition

Second initial condition

The general solution to the differential equation is therefore

                    

Plot[edit]

Below is a plot of this solution and the solution to in the example on p.7-3

our solution (shown in red)

example on p.7-3 (shown in blue)

Plot3 1.jpg

Egm4313.s12.team11.imponenti 22:31, 20 February 2012 (UTC)