University of Florida/Egm4313/s12.team11.imponenti/R6.4

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R6.4[edit | edit source]

solved by Luca Imponenti

Problem Statement[edit | edit source]

Consider the L2-ODE-CC (5) p.7b-7 with the window function f(x) p.9-8 as excitation:

and the initial conditions

1. Find such that:

with the same initial conditions as above.

Plot for for x in

2. Use the matlab command ode45 to integrate the L2-ODE-CC, and plot the numerical solution to compare with the analytical solution.

Level 1:

Fourier Series[edit | edit source]

One period of the window function p9.8 is described as follows

From the above intervals one can see that the period, and therefore Applying the Euler formulas from to the Fourier coefficients are computed:

The integral from to can be omitted from this point on since it is always zero.


The coefficients give the Fourier series:

Homogeneous Solution[edit | edit source]

Considering the homogeneous case of our ODE:

The characteristic equation is

Therefore our homogeneous solution is of the form

Particular Solution[edit | edit source]

Considering the case with f(x) as excitation

The solution will be of the form

Taking the derivatives

Plugging these back into the ODE:

Setting the two constants equal

This is valid for all values of n. Since the coefficients of the excitation and are zero for all even n, then the coefficients and will also be zero, so we must only find these coefficients for odd n's. Now carrying out the sum to and comparing like terms yields the following sets of equations. Written in matrix form:

Assuming this matrix can be solved to obtain

For the remaining coefficients to be solved all sums will be used so a more general equation may be written:

Results of these calculations are shown below:

The solution to the particular case can be written for all n (assuming A=1):


General Solution[edit | edit source]

The general solution is


Different coefficients will be calculated for each n. These coefficients are easily solved for by applying the given initial conditions. Below are the calculations for n=2.

Applying the first initial condition

Taking the derivative

Applying the second initial condition

Solving the two equations for two unknowns yields:

So the general solution for n=2 is:

Below is a plot showing the general solutions for n=2,4,8:



Matlab Plots[edit | edit source]

Using ode45 the following graph was generated for n=0:


and for n=1