MATH 202 Final, Fall 2002

American University of Beirut MATH 202 Differential Equations Spring 2007 Final Exam Name: ............................... ID #: ................................. circle your section please: 13 (T 11) 14 (T 12:30) 15 (T 2) 16 (T 9:30) 1. (10 points) Find the general solution of y(4) − y = ex 2. (15 points) Use a power series centered at the point x = 0 to find the general solution of: (1 + x3)y′′ − 6xy = 0 (simplify the recurrence as much as possible before solving for ca, c1, . . .) 3. (15 points) Find a) L {(3t + 1)U(t − 1)} b) L−1 � e−2s s2(s − 1) � c) L � t � t 0 τe−τdτ � 4. a. (6 points) Use the substitution x = et to solve the following differential equation in terms of Bessel functions: d2y dt2 + � e2t − 1 4 � y = 0 b. (4 points) express your answer in terms of the elementary functions c. (10 points) Show that y = √xJν(x) is a solution of the differential equation x2y′′ + � x2 − ν2 + 1 4 � y = 0 5. (15 points) Use the Laplace transform to solve the following DE: y′ + y = f(t) , y(0) = 5 where f(t) = � 0 0 ≤ t < π 3 cos t t ≥ π 6. (15 points) Use a generalized (i.e. Frobenius) power series of the form y = +∞ � n=0 cnxn+r to find ONE solution of 3x2y′′ + 2xy′ − xy = 0 corresponding to the LARGER indicial roots 7. (10 points) Find the general solution of the given system      dx dt = x + 2y dy dt = 4x + 3y