MATH 202 Final, Spring 2005

MERICAN UNTVER! . . LIBKARY Math 202-Final Exam (Spring 05) OF SEIRUT e Please write your section number on your booklet. e Please answer each problem on the indicated page(s) of the booklet. Any part of your answer not written on the indicated page(s) will not be graded. e Unjustified answers will receive little or no credit. Problem 1 (answer on pages 1, 2, and 3 of the booklet.) (15 pts each) Solve the following differential equations. dy 4 (a) s--ty=s 5 2 e dt 2 (y? + yz) dz — x? dy =0 Problem 2 (answer on page 4 of the booklet.) (10 pts) Find the general solution of the Bessel equation 1 zy” + zy! + (92" - 3) y=0 on the interval (0,00). Problem 3 (answer on pages 5, 6, and 7 of the booklet.) (38 pts) Find two linearly independent series solutions of the equation cy” +3(1—z)y' + 3y =0 about the regular singular point z = 0. (It is enough to find the first 4 terms of each series. GRR, RRI, and RRII must be clearly stated.) Problem 4 (answer on pages 8 and 9 of the booklet.) (25 pts) Use the Laplace transform to solve the IVP | y" + 4y' + by = 6(t — 27) y(0)=9, y'(0)=0. Problem 5 (answer on pages 10 and 11 of the booklet.) (8 pts each) Let a be a constant. Find the following transforms. (i) L {f re cos(t ~ 7)dr} Gi) L{(at —3)u(t—1)} Problem 6 (answer on pages 12, 13, and 14 of the booklet.) (28 pts) Find the general solution of the system 010 X'=/] -1004X 0 0 3 on the interval (—0o, 00). (Don’t forget to write your answer in both complex and real forms.) Problem 7 (answer on pages 15 and 16 of the booklet.) (10 pts) Prove the following theorem in the case n = 2: Theorem. Let 1, y2,..., Yn be solutions of the nth-order linear homogeneous ODE d"y dy On (2) as a(x)" + ao(z)y = 0 on an interval J where @o(z),a;(x),...,@,(z) are continuous and where a,(z) is never zero. Then Yi; Y2,---,Yn linearly independent on I == W(y1, yo,---, Yn) # 0 everywhere in J.