Final Exam MATH 2552 PDF

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Final exam for math 2552 differential equaitons. with answers...

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Final – 2552, Fall 2020 Instructor: Wenjing Liao

• This is a take-home exam. You can use books, notes and calculators. • This exam is from 8am to 11am on December 3rd. Your solution needs to be submitted to Gradescope before 11am. Late submissions are Not accepted. To ensure a proper submission, you are recommended to scan your work and submit to Gradescope about 10 min before the deadline. You are recommended to scan your solution into the pdf format. If you have problems in uploading the exam, please e-mail your PDF file to your TA and cc your instructor 5 min before the deadline, using your GT e-mail address, with an explanation. • There are different ways you can write your solutions. For example, Method 1: You can print out the test, write on the printed hardcopy, and scan your solutions, or take photos of your solutions. Method 2: You can write your solutions on blank papers, and scan your solutions, or take photos of your solutions. Method 3: You can use an appropriate device (e.g., a tablet) to write directly on the PDF file and save your work. • All the work in your submission has to be handwritten by yourself. Typed texts are not accepted. • Use one solution per page for one problem. Do not solve a single problem on multiple pages. Do not combine solutions of multiple problems on a single page. • Provide detailed solutions, including all steps and computations. Answers without explanations or supporting computations may end up receiving no credit. • Write clearly and neatly. Clearly indicate which part of your solution is your intended answer to the problem, by circling or boxing. Statement: Sign below to acknowledge the following sentence. I commit to uphold the ideals of honor and integrity by refusing to betray the trust bestowed upon me as a member of the Georgia Tech Community. Signature:

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Problem 1 (14 points): Consider the following differential equation: 2y

dy t , y(5) = 2. =√ 2 dt t −9

Please copy what is in the blue box in your solution. An extra 10 min is given for this. (1) Solve this initial value problem. (8 points)

(2) Find the approximate value of the solution of the given initial value problem at t = 5.2 using the Euler method with h = 0.2. Keep four digits. (3 points)

(3) Find the approximate value of the solution of the given initial value problem at t = 5.2 using the improved Euler method with h = 0.2. Keep four digits. (3 points)

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Problem 2 (6 points): For the following equation, determine a suitable form for the particular solution Y (t) if the method of underdetermined coefficients is to be used. No need to solve for the coefficients. Please copy what is in the blue box in your solution. y00 − 3y0 − 4y = t2 et + 2te4t sin t.

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Problem 3 (22 points): Compute the following Laplace transform and inverse Laplace transform. Please copy what is in the blue box in your solution. (1) Compute  ⇢ (s + 2)e2s . L1 (s − 1)(s + 1)2 (7 points)

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(2) Compute L {f }, where f is the following periodic function: (7 points)

Periodic function 2 Please copy what is in the blue box in your solution.

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Compute f (t) = L1 (3)

⇢

1 + eπs 2 (s + 1)(1 − eπs )



.

and graph the function f (t). (8 points) Please copy what is in the blue box in your solution.

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Problem 4 (12 points): Solve the following initial value problem y(3) + 3y00 + 4y0 + 2y = u2 (t) where y(0) = 0, y0 (0) = 0, y 00(0) = 0. Notice that s3 + 3s2 + 4s + 2 = (s + 1)(s2 + 2s + 2). Please copy what is in the blue box in your solution.

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Problem 5: (17 points): A mass that weighs 8 lb stretches a spring 24 in. The system is acted on by an external force of tan(4t) lb at 0 < t < π8 . Suppose the mass is pulled down 6 in. and then released. Notice that 1 ft = 12in. and g = 32 ft/s2 .

Figure 1 (1) Let y(t) be the position of the mass at t. Write down a differential equation for y(t). Do not forget the initial value. (6 points)

continue to the next page

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(2) Find the general solution of the differential equation in Part (1). No need to plug in the initial value to figure out the constants. (11 points) Hint: You might need to use the method of variation of parameters. If this computation is long, please be patient. You might need to use Z 1 sec x = and sec xdx = ln | sec x + tan x| + C. cos x

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Problem 6 (29 points): Consider the nonlinear system of equations: ( dx = (3 + x)(x − y − 1) dt dy = (1 + x + y)(1 − y) dt and answer the following questions. Please copy what is in the blue box in every sub-question (such as Part (1) to Part (7)) of your solution. (1) Show that the critical points of this system are (0, −1), (2, 1), (−3, 1) and (−3, 2). (4 points)

(2) Write down the Jacobian matrix for this system. (4 points)

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(3) Find the corresponding linear system near the critical point (0, −1). Find the eigenvalues this linear system, and draw a phase portrait near (0, −1). You do not need to compute the eigenvectors. (4 points)

(4) Find the corresponding linear system near the critical point (2, 1). Find the eigenvalues and eigenvectors of this linear system. Draw a phase portrait near (2, 1). (4 points)

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(5) Find the corresponding linear system near the critical point (−3, 1). Find the eigenvalues of this linear system. Draw a phase portrait near (−3, 1). (4 points)

(6) Find the corresponding linear system near the critical point (−3, 2). Find the eigenvalues of this linear system. Draw a phase portrait near (−3, 2). (4 points)

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(7) Draw a phase portrait of the nonlinear system, and classify the critical points as asymptotically stable, stable but not asymptotically stable, or unstable critical points. (5 points)...