MATH 202 Final, Spring 2008

Math 202 Final Exam • The duration of the test is 2 hours • No calculators are allowed NAME:————————————————— SECTION:————— 1. (15 points) Solve the following differential equation (x + 2)sinydx + xcosydy = 0. 1 - 2 2. (15 points) Consider the differential equation y′′ − 5y′ + 6y = e3t(5t2 + t + 1). Solve by first substituting y = e3tu and then use the method of undetermined coefficients. 3 - 4 3. (15 points) Determine the singular points of the given differential equation and show that it is regular. Solve the differential equation using the power series solution using �∞ n=0 cnxn+r by finding first three terms of each solution. xy′′ + y′ + y = 0 5 - 6 4. (15 points) Use Laplace transform to solve the following initial value problem. y′′ + 2y′ + 2y = u(t − 5) − u(t − 20), y(0) = 0, y′(0) = 0. 7 - 8 5. (15 points) Use Laplace transform to find a solution of the following initial value problem y′′ + 4y = g(t), y(0) = 3, y′(0) = −1 where g(t) is a continuous function of t. 9 - 10 6. (15 points) Use the method of undetermined coefficients to solve the following system of equations dx dt = −4x + y + z + 3 dy dt = x + 5y + −z + 5 dz dt = y − 3z − 2 11 - 12 7. (10 points) Let S be the portion of the plane x + y + z = 1 that lies in the first octant, and let C be the boundary of S, traversed counterclockwise. Calculate � C F.dr where F = −3 2y2i − 2xyj + yzk. 13 - 14