University of Florida/Egm4313/s12.team11.perez.gp/R5.9

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R5.9[edit | edit source]

Problem Statement[edit | edit source]

Consider the L2-ODE-CC (5) p.7b-7 with as excitation:

(5) p.7b-7

(1) p.7c-28

and the initial conditions

.

Part 1[edit | edit source]

Project the excitation on the polynomial basis

(1)

i.e., find such that

(2)

for x in , and for n = 3, 6, 9.

Plot and to show uniform approximation and convergence.

Note that:

(3)

Solution[edit | edit source]

Using Matlab, this is the code that was used to produce the results:

Part 2[edit | edit source]

Find such that:

(1) p.7c-27

with the same initial conditions as in (2) p.7c-28.

Plot for n = 3, 6, 9, for x in .

In a series of separate plots, compare the results obtained with the projected excitation on polynomial basis to those with truncated Taylor series of the excitation. Plot also the numerical solution as a baseline for comparison.

Solution[edit | edit source]




Using integration by parts, and then with the help of of

General Binomial Theorem



Solution[edit | edit source]

For :



For substitution by parts,







Therefore:

                                     

Using the General Binomial Theorem:



Therefore:

Which we have previously found that answer as:

                                     




For :



Initially we use the following substitutions:



First let us consider the first term:

Next, we use the integration by parts:






Next let us consider the second term:

Again, we will use integration by parts:






Therefore:





Re-substituting for t:







Therefore:

                                     



Using the General Binomial Theorem for the integral with t substitution :



Therefore:

Which we have previously found that answer as: