University of Florida/Egm4313/IEA-f13-team10/R6

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Report 6[edit]

Problem 1[edit]

Problem Statement[edit]

ODE:

Part 1: show that cos7x and sin7x are linearly independent using the Wronskian and Gramian.

Part 2: Find 2 equations for the two unknowns M, N, and solve for M, N.

Part 3: Find the overall solution y(x) that corresponds to the initial conditions:

Plot the solution over 3 periods

Solution[edit]

Part 1[edit]

Wronskian: Function is linearly independent if








g(x) and f(x) are linearly independent

Gramian: Function is linearly independent if











g(x) and f(x) are linearly independent

Part 2[edit]

The particular solution for a will be:



Differentiate to get:





Plug the derivatives into the equation:





Separate the sin and cos terms to get 2 equations in order to solve for M and N





dividing each equation by cos7x and sin7x respectively:









So the particular solution is:



Part 3[edit]

The overall solution can be found by:



The roots given in the problem statement

Lead to the homogeneous solution of:



Combining the homogeneous and particular solution gives us:



Solving for the constants by using the initial conditions









The overall solution is:



Plot[edit]

Plot

over 3 periods:

R6.1.PNG

Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 2[edit]

Problem Statement[edit]

Complete the solution to problem on p.8-6.
Find the overall solution
that corresponds to the initial condition
Plot solution over 3 periods.

Solution[edit]

Given:













Solve for M and N:




Using initial conditions given find A and B

After applying initial conditions, we get





Plot 1.PNG
Plot 2.PNG

Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 3[edit]

Problem Statement[edit]

Is the given function even or odd or neither even nor odd? Find its Fourier Series.

Solution[edit]



so is an even function.

The Fourier series is .



For



For


The above integral requires two iterations of integration by parts. Which gives

Similarly, integration by parts needs to be used twice to solve the following integral.


So the Fourier series for is


Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 4[edit]

Problem Statement[edit]

1) Develop the Fourier series of. Plotand develop the truncated Fourier series.

for n = 0,1,2,4,8. Observe the values of at the points of discontinuities, and the Gibbs phenomenon. Transform the variable so to obtain the Fourier series expansion of . Level 1: n=0,1.

2)Do the same as above, but usingto obtain the Fourier series expansion of; compare to the result obtained above. Level 1: n=0,1.

Solution[edit]

Part 1[edit]

To begin, the function was determined to be even. Even functions reduce to a cosine Fourier series. Because , has a period of 4, the length is 2.











For n=0,


For n=1,






Plot (A=1)

R6 p4(1).JPG

Part 2[edit]

To begin, the function was determined to be odd. Even functions reduce to a sine Fourier series. Because , has a period of 4, the length is 2.







from 0 to 4





from 0 to 4





For n=0,


For n=1,





Plot (A=1)

R6 p4(2).JPG

Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 5[edit]

Problem Statement[edit]

Find the separated ODE's for the Heat Equation:

(1)

heat capacity

Solution[edit]

Separation of Variables:

Assume:

(2)

(3)

(4)

(5)

Plug (2) and (3) into Heat Equation (1):

(6)

Rearrange (6) to combine like terms:















Solution:

Separated ODE's for Heat Equation:





Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 6[edit]

Problem Statement[edit]

Verify (4)-(5) p.19-9
(4) for
(5) for

Solution[edit]

Verification of (4)[edit]

Using the integral scalar product calculation,


Substituting in sin values,


Using and

You can substitute z into the integral instead of x.


Integrating,
from to
Since , the equation with its sin values turns into 0-0=0


Verification of (5)[edit]

You can use the same equation from the verification of (4) from this point:
from to
Putting those values in and substituting L back in the equation, it turns into

Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.

Problem 7[edit]

Problem Statement[edit]


Plot the truncated series for n=5.


Solution[edit]





C=3 and L=2
Plot

P7.1.JPG

Plot

P7.2.JPG

Plot

P7.3.JPG

Plot

P7.4.JPG

Honor Pledge[edit]

On our honor, we solved this problem on our own, without aid from online solutions or solutions from previous semesters.