MAE105
Final Exam
(open book, closed notes, no phone, calculators or computers)
Name:_________________________
Time: 3:00 to 6:00pm
Date: December 10, 2004
Problem 1
(a) (1 Point) Find a general expression
x
=
x
(
t
), for the characteristics of the following PDE:
∂
t
∂
u
+
x
cos
t
∂
x
∂
u
=
t u .
(1)
[Note that your expression must include a constant of integration, say,
x
(0)
=
x
0
.]
(b) (0.5 Point) In the
x
,
t
plane,
t >
0, sketch a typical characteristic curve for 0
≤
t <
2
π
.
(c) (1 Point) Show that on the characteristics, ODE (1) reduces to
dt
du
=
t u .
(2)
Find the general solution of this PDE.
(d) (1 Point) Specialize the solution in (c) such that at
t
=
0, we have
u
(
x
0
,0)
=
u
0
=
(ln(
x
0
)
+
1).
Problem 2
Consider the wave equation
∂
t
2
∂
2
u

∂
x
2
∂
2
u
=
0
(3)
in a finite domain, 0
< x <
1,
t >
0, with the boundary conditions
u
(0,
t
)
=
u
(1,
t
)
=
0 ,
(4)
and initial conditions
u
(
x
, 0)
=
f
(
x
)
=
1

x
x
sin(
π
x
)
for
1
/
2
< x <
1
.
for
0
< x <
1
/
2
(5)
∂
t
∂
u
(
x
, 0)
=
g
(
x
)
=
sin(
π
x
) ,
0
< x <
1
.
(a) (1 Point) Draw the (
x
,
t
)plane, and below the
x
axis, draw the initial conditions in two
graphs, as discussed in the class and in your book.
(b) (1 Point) Extend the initial conditions such that the boundary conditions are satisfied for all
t >
0, if we use the general solution for the infinite domain.