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Eric Essenwein


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Problem R6.4[edit]


Consider the L2-ODE-CC (5) p7b-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 (4.1).
Plot for n=2,4,8, for x in [0,10].
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:n=0,1

Part 1 Solution[edit]

First, we shift the excitation f(x) to the left by introducing a new independent variable, t. This allows the period to start at zero.


The piecewise representation of the window function is now (in terms of t) as follows:



To find the Fourier transform, the period of oscillation is determined:


And the frequency of oscillation:


The general form of a Fourier transform is


The following equations are given values of the constants in (4.7), evaluated for the function in (4.4):




In their simplest forms:



Plugging in r(x)=f(t) in (4.0),


From the general form, a particular solution to (4.13) and its derivatives are as follows:




By plugging in (4.14), (4.15), and (4.16) into (4.13), we find An and Bn in terms of another constant, Cn:




After substituting t with (4.3), the solutions are shown for n=2,4,8.





Part 2 Solution[edit]

The excitation for n=0 and n=1 are the same, because .

The homogeneous solution to (4.0) is


For , the particular solution is


Evaluating, we find that


Thus, the complete solution is


Integrating using MATLAB's ode45 command, the following plot is obtained for y:


Solved and Typed By -
Reviewed By -

Problem R5.6[edit]


Consider the following L2-ODE-CC; see p.6-6:


Homogeneous solution:


Particular solution:


Complete the solution for this problem.

Find the overall solution that corresponds to the initial condition (3b) p.3-7



Start by finding and .



Substitute and its derivatives into (6.0) to find and .


Separating terms and setting equal to the excitation from (6.0):


From (6.7), we solve coefficients to get


Solving for and :


Which gives us the particular solution:


For the general solution,




To solve for and , we use initial conditions from (6.3):


Which simplifies to:


For the second initial condition from (6.3):





We can now write with all coefficients known.


Final Equation




Solved and Typed By - Egm4313.s12.team1.essenwein (talk) 23:45, 19 March 2012 (UTC)
Reviewed By -