• University of Houston

• MATH 3331

Math 3331 Exam 2. Sanders Spring 2023 This exam has five problems, and all five will be graded. Use my supplied paper only. Return your solution sheets with the problems in order. Put your name, last name first, and student id number on each solution sheet you turn in. Each problem is worth 20 points with parts equally weighted unless indicated otherwise. 1. Factorize the following second order differential operators into the composition of two first order operators. (a) d 2u dx2 − 4u. (b) d 2u dx2 + du dx − 2u. 2. Determine the general solution (homogeneous solution + particular solution) to each of the following by using the method of guessing. (a) d 2u dx2 − du dx = e 2x . (c) d 2u dx2 − du dx = e x . (b) d 2u dx2 − du dx = x + 2. (d) d 2u dx2 − du dx = sin(x). (Since the LHs are all the same, you only need to find the homogeneous solution once.) 3. Use Duhamel to find the solution of each of the following initial value problems. (a) d 2u dx2 − u = e x , u(0) = ux(0) = 0. (b) d 2u dx2 = x, u(0) = ux(0) = 0. 4. Write each of the following scal