Solved by: Andrea Vargas
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
1. To obtain the equations for the coefficients we use the following equation from (1) p7-11:
![{\displaystyle \sum _{j=0}^{3}[c_{j+2}(j+2)(j+1)-3c_{j+1}(j+1)+2c_{j}]x^{j}-3c_{5}(5)x^{4}+2[c_{4}x^{4}+c^{5}x^{5}]=4x^{2}-6x^{5}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66548e831aed1df2c0af03f920d318c9ff821faf)
Finding the coefficients of 
where 
:
![{\displaystyle [c_{3}(3)(2)-3(c_{2})(2)+2c_{1}]x^{1}=[6c_{3}-6c_{2}+2c_{1}]x\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/588b04bbf17c37133fad9347ca49ae12d1e4f135)
Finding the coefficients of 
where 
:
![{\displaystyle [c_{4}(4)(3)-3(c_{3})(3)+2c_{2}]x^{2}=[12c_{4}-9c_{3}+2c_{2}]x^{2}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9977462a0e7926e2648ec2ec9a0880243279768)
Finding the coefficients of 
where 
:
![{\displaystyle [c_{5}(5)(4)-3(c_{4})(4)+2c_{3}]x^{3}=[20c_{4}-12c_{3}+2c_{2}]x^{3}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a721f2df736c522657a58a8a965ad35f0ead0d79)
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:
This yields the particular solution:
Homogeneous Solution
edit
Then, we can find the characteristic equation:
Then the solution for the homogeneous equation becomes:
Using the given initial conditions we find the overall solution:
Using the initial conditions to solve for 
and 
The general solution becomes
Below is a plot of the solution:
--Egm4313.s12.team11.vargas.aa 05:56, 21 February 2012 (UTC)
