Midterm 421_2 solution

MATH 421:01 - Midterm Exam 2 - November 14th 2023 80 minutes Name: NetID: (e.g., ftc12 ) On my honor, I have neither received nor given any unauthorized assistance on this examination. Signature Instructions 1. This exam is closed book. Calculators, electronic devices, notes, books, formula sheets, and other outside materials are not allowed, except a cheat sheet on one side of a letter-size paper. Phones must be turned o ↵ and put away in your backpacks. 2. Show all your work. Answers must be justified using techniques that have been taught thus far in this course, and an answer without such justification only worths 20% of the points. 3. Unless otherwise stated, you are expected to simiplify your final answer. 4. Do not detach any of the pages. It must come back as part of the stapled package you return to us when you turn in the exam, or we will deduct two (2) points from your score. 5. Write your work and answer in the space below a given problem and the designated pages of that problem. E.g., if you are answering question #1, do not write it in the space under question #2 or the other designated pages of question #2. The software we use to sort the pages for grading will only locate your answer if it is in the space of the corresponding question. 6. The exam has 5 questions and will be scored out of 40 points.

Version A Math 421 Midterm Exam 2 Page 2 1. (2 pts) Classify the given partial di ↵ erential equations as hyperbolic, parobolic or elliptic. @ 2 u @ x 2 + 2 @ 2 u @ x @ y + 2 @ 2 u @ y 2 = 0 , @ 2 u @ x 2 + 3 @ 2 u @ y 2 + 5 @ 2 u @ x @ y + 7 @ u @ x + 9 @ u @ y = 11 .

Version A Math 421 Midterm Exam 2 Page 5 3. (7 pts) Find the complex Fourier series of f ( x ) on the interval [ - ⇡ , ⇡ ] f ( x ) = e 2 x + x.

Version A Math 421 Midterm Exam 2 Page 7 4. (12 pts) Solve the following boundary-value problem for heat equation k @ 2 u @ x 2 = @ u @ t , 0 < x < ⇡ , t > 0 , @ u @ x (0 , t ) = 0 , @ u @ x ( ⇡ , t ) = 0 , t > 0 , u ( x, 0) = cos 2 x, 0 < x < ⇡ .

Version A Math 421 Midterm Exam 2 Page 10 5. (12 pts) Solve the following boundary-value problem for Laplace equation @ 2 u @ x 2 + @ 2 u @ y 2 = 0 , 0 < x < 1 , 0 < y < 1 , u (0 , y ) = 0 , u (1 , y ) = 0 , 0 < y < 1 , u ( x, 0) = 0 , @ u @ y ( x, 1) = x (1 - x ) , 0 < x < 1 .

Version A Math 421 Midterm Exam 2 Page 11 (Continue to answer Question 5 on this page.)