MAE105
Midterm 2
(open book, closed notes)
Name:_________________________
Time: 3:35 to 4:45pm
Date: May 22, 2007
Problem 1
(a) (1 Point) Find a general expression
x
=
x
(
t
), for the characteristics of the following PDE:
∂
u
∂
t

x
cos
t
∂
u
∂
x
= 
u
sin
t
.
[Note that your expression must include a constant of integration, say,
x
(0)
=
x
0
.]
(b) (1 Point) In the
x
,
t
plane,
t
> 0, sketch a typical characteristic curve.
(c) (1 Point) Find the general solution of this PDE.
(d) (1 Point) Specialize the solution in (b) such that at
t
=
0, we have
u
(
x
0
,0)
=
u
0
=
(
x
2
0
+
1).
Problem 2
Consider the wav e
equation
∂
2
u
∂
t
2

∂
2
u
∂
x
2
=
0
(1)
in an infinite domain,

∞
<
x
<
∞
,
t
> 0, with the initial conditions
u
(
x
, 0)
=
f
(
x
)
=
x
1

x
for
0 <
x
<
1
/ 2
for
1 / 2 <
x
<
1
,
∂
u
∂
t
(
x
, 0)
=
g
(
x
)
=
sin(
π
x
) ,
0 <
x
<
1
,
with both
f
(
x
) and
g
(
x
) being zero for all other values of
x
.
(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) (1.5 Points) In the
x
,
t
plane, draw the characteristics that pass through the following points:
[
x
=
0,
t
=
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 NeimanNassat
 Boundary value problem, Partial differential equation, general solution, Boundary conditions

Click to edit the document details