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MATH 3326 FINAL EXAMINATION
SPRING SEMESTER 2009
Lance L. Littlejohn
Name SCLW"nDN.$.
Instructions: Show all work. Partial credit can only be given if sufficient work accompa
nies each answer. Calculators may be used but exact answers are required. This examination
is out of 70 points. GOOD LUCK!
Problem No.
L
2.
3.
4.
5.
6.
Grade
L (10 POINTS) Using d'Alembert's formula, solve the Cauchy problem
Utt - c
2
u
xx
(-00 < X < 00, t> 0)
u(x,O) - cos x, 11t(X,0) = e-
x
(-00 < x < 00)
with c = 4. Simplify your answer as much as possible.
\ Y\ --It \ s Q.o,:;,e.,~ 'f l~) "" c.'Os X ~ "-f !x)-=- e.::-'"
d\M~\1f1~t\s. Se.\,vAiovv in fu0 Q.C<~Q-\~~Ud::;
{,{u<;c):::. '{?\X,+4-b)+'f(X-'-tt)+-t-~ e-Ucl-v-.-
~ ~-4-t
Points
/70
2. Consider the first-order linear PDE
U
x
+ xU
y
+ 3u = 2. (1)
(a) (3 POINTS) Find, and solve, the characteristic equation associated with (1).
SO ~= ~~'-4-C ~ \t;--t"= ~J
(b) (4 POINTS) Using ~(x,y) = x, find a transformation 1) ...

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