University of Florida/Egm4313/s12 Report 4, Problem 4.3

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Problem 4.3[edit]

Problem 4.3 Part 1[edit]

Problem Statement[edit]

Solution[edit]

Problem 4.3 Part 2[edit]

Problem Statement[edit]

Given:

where

With initial conditions:

Find the overall solution for and plot these solutions on the interval from

Solution[edit]

First we find the homogeneous solution to the ODE:
The characteristic equation is:


Then,
Therefore the homogeneous solution is:


Now to find the particulate solution
For n=4



We can then use a matrix to organize the known coefficients:



Then, using MATLAB and the backlash operator we can solve for these unknowns:
Therefore


Superposing the homogeneous and particulate solution we get


Differentiating:
Evaluating at the initial conditions:



We obtain:



Finally we have:


For n=7



We can then use a matrix to organize the known coefficients:



Then, using MATLAB and the backlash operator we can solve for these unknowns:
Therefore


Superposing the homogeneous and particulate solution we get


Differentiating:
Evaluating at the initial conditions:



We obtain: