• University of Houston

• MATH 3331

Math 3331 Exam 2. Sanders Fall 2020 This exam has five problems each worth 20 points with parts equally weighted unless indicated otherwise. Your time limit is one hour and twenty minutes. No notes or calculators allowed. Do not communicate with others during the exam. (1) Please use lineless paper (i.e. Xerox paper) if you can. (2) Add a cover sheet to the front of your exam solutions with your name (last name first), student id number, your signature and your student photo id card. (3) Scan to pdf the cover sheet followed by solutions in numerical order. 1. Determine the general solution (homogeneous + particular solution) to each of the following by using the method of guessing. (a) d 2u dx2 − u = sin x. (c) d 2u dx2 − du dx = e 2x . (b) d 2u dx2 − u = e x . (d) d 2u dx2 − du dx = x. 2. Solve the following inhomogeneous IVPs using Duhamel’s principle. (a) d 2u dx2 + u = x, u(0) = u ′ (0) = 0. (b) d 2u dx2 − u = x, u(0) = u ′ (0) = 0. Half credit will be assigned for properly setting up Duhamel’s integral and the other half for integrating properly. You’ll need to use integration by parts. 3. Write the following scalar differential equations as first order systems. (a) d 2u dt2 − u = 0. (c) d 2u dt2 = u du dt + sin(u). (b) d 2u dt2 − 2 du dt + u = 0. (d) d 3u dt3 − du dt = tu. 4. Find all eigenvalues and corresponding eigenvectors for the following. (a)  0 1 −2 3  (b)  −5 6 −4 5  5. You may freely use the following similarity transformation. R −1AR ≡  1 −1 −1 2   −1 2 −1 2   2 1 1 1  =  0 0 0 1  ≡ Λ Please use what’s given. There’s no need to calculate eigenvalues or eigenvectors here. (a) Determine e At where A =  −1 2 −1 2  . (b) Now, use this to solve the initial value pro