University of Florida/Egm4313/IEA-f13-team10/R3

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Report 3[edit]

Problem 1[edit]

Problem Statement[edit]

Find the complete homogeneous solution using variation of parameters

Solution[edit]



The solution is
Therefore, and

Plugging this back into the original homogeneous equation,

so




Checking the answer








Plugging this into the original homogeneous equation


Plugging in values for y and its derivatives, everything cancels out to zero.


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]

Find and plot the solution for the L2-ODE_CC

Solution[edit]



This is a linear, first order ODE with constant coefficients.

To find the general solution to this ODE set

so that and

Substituting in y to the ODE and factoring out we get:


Using the quadratic formula to solve for r we get

where and

Solving to get

Since we have a repeated root, we need to find v(x) so that y2(x) = v(x)y1(x)

Taking the first and second derivative of y2(x) we get:




Substituting into the original ODE, we get:


solving for v(x) = 0 so v(x) = kx + c

So y2(x) = x y1(x)


We get the general solution

Now with the initial values y(0) = 1 and y'(0) = 0






,


2.3graph.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]

Problem Sec 2.4 problem 3[edit]

How does the frequency of the harmonic oscillation change if we (i) double the mass (ii) take a spring of twice the modulus?

Problem Sec 2.4 problem 4[edit]

Could you make a harmonic oscillation move faster by giving the body a greater push?

Solution[edit]

Problem Sec 2.4 problem 3[edit]
Part 1[edit]



Now double the mass


The frequency is decreased by .

Part 2[edit]

Multiply k by 2

The frequency is increased by .

Problem Sec 2.4 problem 4[edit]

No because frequency depends on the ratio of the spring modulus and mass.

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]

Section 2.4 Problem 16[edit]

Show the maxima of an underdamped motion occur at equidistant t-values and find the distance.

Section 2.4 Problem 17[edit]

Determine the values of t corresponding to the maxima and minima of the oscillation . Check your result by graphing y(t).

Solution[edit]

Section 2.4 Problem 16[edit]

Part 1=[edit]

The general solution of underdamped motion is

The maximas occur at
Set the two equations equal to each other a solve for t.



where n=0,1,2,3.....

Part 2[edit]

shows delta is a constant.
The periodic distance between maximas is

Section 2.4 Problem 17[edit]




To find critical points, set y'(t)=0







where n=0,1,2,3...
2.4.17.JPG
As seen in the graph, the maximum of t was at and the minimum was at .

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]

Using the formula for Taylor series at x = 0 (the origin, i.e., McLaurin series), develop into Taylor series at the origin x = 0 for the following functions: cos x, sin x, exp(x), tan x, and write these series in compact form with the summation sign and a single summand.

Solution[edit]

Part 1
cos x
Taylor Series:

when








n = 0, 1, 2, 3 ... N

2n = 0, 2, 4, 6, ... 2N


Part 2
sin x
Taylor Series:

when









Part 3
exp x
Taylor Series:

when









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]

Part 1
Find and plot the solution for the L2-ODE-CC

Initial conditions: y(0) = 1, y'(0) = 0
No excitation: r(x) = 0

Part 2
In another Fig., superpose 3 Figs.: (a) this Fig.,
(b) the Fig. in R2.6 P. 5-6, (c) the Fig. in R2.1 P. 3-7.

Solution[edit]

Part 1

The characteristic equation of the given ODE is:


Using the quadratic formula to solve for

where

Solving to get

,

Therefore, the general solution of the given ODE is

Now we solve for , using the given initial conditions

We have

Substituting into ,

We get,




We have

Differentiating , we get:


.

Substituting into , we get:






Therefore, we get

Hence the solution of the given ODE is

Fig. 1:

Problem6 Part1.jpg

Part 2




Fig. 2:

Problem6 Part1.jpg

Fig. 3:

Problem6 Part2 2.jpg

Fig. 4:

Problem6 Part2 3.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 7[edit]

Problem Statement[edit]

Consider the same system as in the Example p.7-3, i.e., the same L2-ODE-CC (4) p.5-5 and initial condi- (2) p.3-4, but with the following excitation:

Solution[edit]





Replacing with and after simplifying we get,



The root here is . So we can solve for our constants,






Using the initial conditions y(0)=4 and y'(0)=-5 we can solve for our constants,




So the solution 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 8[edit]

Problem Statement[edit]

Plot the error between the exact derivative and the approximate derivative, i.e.

from



For ε = .0001, .0003, .0006, and .001 and λ =.3

Solution[edit]

Since the above equation is the error between the exact derivative and the approximate derivative, it must be plotted with the correct values of \epsilon and \lambda, from x = -15 to 15

Case 1[edit]

ε=.0001
Pic1.png

Case 2[edit]

ε=.0003
Pic2.png

Case 3[edit]

ε=.0006
Pic3.png

Case 4[edit]

ε=.001
Pic4.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 9[edit]

Problem Statement[edit]

Find the complete solution for , with the initial conditions
,
plot the solution y(x)

Solution[edit]

Particular Solution



















Homogeneous Solution




Initial conditions










General Solution

R3plots.PNG

Honor Pledge[edit]

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