MATH 202 Exam 2, Spring 2014

American University of Beirut MATH 202 Differential Equations Spring 2014 Quiz #2 Time: 70 minutes Exercise 1 (22 pts) Find the right form of yp -do not find the constants- for each of the following differential equations a) y’’’ – 3y’’ + 4y’ = 1 + xex + 2ex sin x b) y (5) + 27y’’ = 6x + x2 sin(3x) Exercise 2 (20 pts) Solve the differential equation: y’’ – 2y’ + 2y = ex tan x Exercise 3 (10 pts) Give the general solution of the differential equation: x2y’’’ – 3xy’’ + y’ = 0 Exercise 4 (16 pts) a) Prove that the reduction of order in general for y’’ + p(x) y’ + q(x) y = 0 via the substitution y=y1u(x), where y1 is a non-zero solution of the homogeneous equation gives u’’ + (p(x) + )u’ = 0 b) Use part a) to find the general solution of the differential equation (E) given that y=x3 is a solution (E): (x3 - x2) y’’ +2(-x2 + 2x) y’ -6y = 0 Exercise 5 (24 pts) Consider the differential equation: (E): (x2– 1) y’’ + xy’ – y = 0 Let y = ∑ be a series solution for (E) about the ordinary point x = 0 a) What would be the radius of convergence of the above series? Justify your answer b) Find two series solution for the equation (E) c) Use reduction of order to express the series found in part b) in terms of an integral of the usual functions Exercise 6 (8 pts) Use the substitution 2x + 1 = et to transform the given differential into equation with constant coefficients and then STOP (2x + 1)2y’’ - (4x – 2) y’ + 10y = ln(2x+1)