241 Rimmer - Spring 2007
Midterm #1
1. 100 , 0 1, 0
xx t
u u x t
= < < >
2
2. 0 , 0
xx tt
a u u x L t
= < < >
(
)
(
)
( ) ( )
0, 0, , 0, 0
,0 , ,0 0, 0
x x
t
u t u L t t
u x x u x x L
= = >
= = < <
3. 0, 0 , 0
xx yy
u u y x
π
+ = < < >
(
)
(
)
( )
( ) ( )
( )
( )
,0 0, , 0 0
0, , 0
, is bounded as lim , 0
0 and 0 lead to the trivial solution, skip these
since the boundary is semi-infinite, don
't use hyperbolic solutions
when you ha
x
u x u x x
u y y y
u x y x u x y
π
π
λ λ
→∞
= = >
= < <
→ ∞ =
= <
ve real distinct roots, use the exponential version
4. Use separation of variables to find product
solutions for the given partial differen
tial equation.
(
)
(
)
( ) ( ) ( )
( )
0, 0, 1, 0, 0
,0 sin 2 5sin 5 , 0 1.
0 and 0 lead to the trivial solution, sk
ip these
u t u t t
u x x x x
π π
λ λ
= = >
= − ≤ ≤
= <
(
)
0,
u y y
=
(
)
, 0
as
u x y
x
→
→ ∞
(
)
, 0
u x
π
=
(
)
, 0 0
u x
=
0
xx yy
u u
+ =
π
0
xy
yu u
+ =
15 points
each
