MATH 202 Final, Fall 2001
American University of Beirut · Differential Equations · 1 page
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Math 202 OF BEIRUT
Final Examination A, February 12. 2002
F. Abi-Khuzam
. (20 points)
(v22)(7-9) *
(b) (10 points) Solve the integral equation: y(t) = e7 — ar + fede.
(a) (10 points) Find the inverse Laplace transform of
. (30 points) Use Laplace transforms to solve the problem:
y" 46 + iy = Ur- 2) + d(t- 4), yO) = 0,¥'(0) = L.
Here 8 is the Dirac delta-function, and U is the unit step function.
. (30 points)
a. (15 points) Find all eigenvalues and corresponding eigenvectors for the matrix
1 4.3
A=| 001 |X x *
02 1
Verify your answers by showing they satisfy the definitions of eigenvalue and
eigenvector.
x= yt Avt+ 32
b. (15 points) Find the solution of the problem y' = 7 z where
Z= Wt 2
(0) = 1,¥(0) = 0.2(0) = 1.
. (15 points) Solve the equation: = = y(xy* - 1).
. (25 points) Find the general solution of the differential equation:
yO) ey = yO) 4 Ty" — 20y' + 12y = 16+ 126.
. (30 points)
a. (20 points) Let y = yn c,x" be a solution of the differential equation
y' = Sty = 3°, satisfying y(0) = 1. Find c, asa function of n, then find y and
express it, as far as possible, in closed form.
b. ( 10 points) Solve the differential equation in part (a) directly, without the use of
power series.
. (10 points) If J) and Jo are Bessel functions of the first kind of orders | and 0 respectively,
prove that (xJ;(x))' = avo(s).