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

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:

 

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