University of Florida/Egm4313/s12 Report 3, Problem 3.5

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Problem 3.5[edit]

Solved by: Andrea Vargas

Problem Statement[edit]

Given

1. Obtain the coefficients of
2.Verify all equations by long-hand expansion. Use the series before adjusting the indexes.

3. Put the system of equations in an upper triangular matrix.

4. Solve for by using back substitution.

5. Using the initial conditions find and plot it

Solution[edit]

1. To obtain the equations for the coefficients we use the following equation from (1) p7-11:


Finding the coefficients of where :


Finding the coefficients of where :


Finding the coefficients of where :


Finding the coefficients of where :



2. To verify all the above equations by long hand expansion we use the following equation from (4) p7-12:

Finding the coefficients when :


Finding the coefficients when :


Finding the coefficients when :


Finding the coefficients when :



Finding the coefficients when :



By collecting these terms we can compare them to the equations of part 1.

Coefficients of :


Coefficients of :


Coefficients of :


Coefficients of :



We can see that we obtain the same system of equations to solve for the coefficients with both methods.

3.Constructing the coefficients matrix:

Then, the system becomes:



4. Solving for the coefficients:







Particular solution[edit]

This yields the particular solution:

                                                            

Homogeneous Solution[edit]



Then, we can find the characteristic equation:


Then the solution for the homogeneous equation becomes:

                                                                                      

General Solution[edit]

Using the given initial conditions we find the overall solution:




Using the initial conditions to solve for and





The general solution becomes

                                                     

Plot[edit]

Below is a plot of the solution:

R3 5.jpg

--Egm4313.s12.team11.vargas.aa 05:56, 21 February 2012 (UTC)