MA501 Practice Final Exam March 27, 2017 Please, show your work. This way I might be able to give you a partial credit even if the result is wrong. Problem 1 (25 pt.) Consider linear differential equation x2 y 4xy + 6y = x4 ex , x>0 a) find two linearly independent solutions of the associated linear homogeneous equation. b) find the general solution of the original nonhomogeneous differential equation. Problem 2 (25 pt.) Consider the Sturm - Liouville problem y + 9y = 0; y(5) = y(0); y (5) = y (0) a) classify the problem as regular, periodic or singular b) state the relevant interval c) find the eigenvalues and corresponding eigenfunctions. Problem 3 (25 pt) Solve the Cauchy problem 2y t2 2 y t = 25 x 2 + x e y(x, 0) = cos(x), y t (x, 0) for < x < , t > 0 = cos(2x) for < x < Problem 4 (25 pt.) Solve the boundary value problem u t 2 = 16 xu2 for 0 < x < 1, t > 0 u(0, t) = 2, u(1, t) = 5 u(x, 0) = x2 for t 0 for 0 x 1 1 \f\f\f\f\f\\a a cw\" + w a , in \"U!\" = QM) and '(H'lm = wot) The 36ml 364630\" Is cache\" bx: 1m H: qmangu cu m 7}\" 1C. x-C-l- * 1+CH-M .. +-'- I 3 Hwduds. \"M 9-C o mu 4.) -.= cos 11-59: cos 15!- 2. :1 05(1). (:05 [SM