Math 222 FINAL EXAM, December 14, 2007
Read each problem carefully. Show all your work for each problem. No Calculators!
1. (a) (10) Solve the initial value problem X.
wœ\ß\Ð!Ñœ
"" "
#$ "
Œ Œ
(b) (10) Find all eigenvalues and eigenfunctions of the boundary value
problem: 0 , LC Cœ CÐ!Ñœ!ßCÐ Ñœ!
ww w w
-
2 (a) (6) Sketch the odd periodic extension of period of :Þ#0ÐBÑ
0ÐBÑ œ Ò ß Ó
œ"ß Î#Ÿ
!ß !Ÿ Î#
1x1
x1 (Sketch over the interval 3 3 )
(b (10) Find the Fourier Sine series of x with period 2.Ñ0ÐBÑœ"Bß!Ÿ"ß
$. Find the inverse Laplace Transform of the following functions:
(a) (8) JÐ=Ñ œ #="
=#=&
#
(b) (8) KÐ=Ñ œ Ð"=Ñ/
Ð="Ñ
#=
$
4. (a) 8 Find the solution of the initial value problemÐÑ CÐ>Ñ
, C C œ Ð> # Ñ ? Ð>Ñ CÐ!Ñ œ !ß C Ð!Ñ œ !
ww w
$1!
$1
(b (4) Calculate ÑCÐÑ
51
#
(c) (4) Find the value of such that !CÐ Ñ œ !Þ
(
#
1
5. State if the following IVP's are linear or nonlinear, and solve the problems:
(a) ÐÑ Ñœ!8>C #Cœ ß>!ßCÐ
w-9=Ð>Ñ
>1
(b) ÐÑ8 Cœ ßCÐ!Ñœ#
w#B
CB C
#
'Þ Ð Ñ(a) 8 Determine the appropriate form of a particular solution for theC:
differential equation: .C 'C "$C œ / =38Ð#BÑ B -9=Ð$BÑ
ww w $B #
( .) evaluate the constants inDO NOT C:
(b) 8 Given that is a solution of the differential equationÐÑ C œB
"#
B C $BC %C œ!ß B
#ww w !ß
use reduction of order to find the second linearly independent solution.
