AM351 Spring 08
Due Monday, June 9
Assignment 1
Instructions:
write your solutions clearly, explain your steps carefully and refer to the
orems and lemmas when needed. The assignment is due either in class or in the as
signments drop box (4th floor) by 2pm.
1. (
3 marks
) Consider the following initial value problem involving Legendre’s
equation
(1

x
2
)
y
”

2
xy
+
α
(
α
+ 1)
y
= 0
,
y
(0) = 1
, y
(0) = 5
.
On what interval does the existence and uniqueness theorem stated in class pre
dict that a unique solution to the problem will exist? Justify your answer.
2. (
3 marks
)
(a) Use the definition to show that the functions
f
1
(
x
) = 1
,
f
2
(
x
) =
x,
f
3
(
x
) =
x
2
are linearly independent on the real line.
(b) The above result can be generalized to show that the set of functions
f
1
(
x
) = 1
,
f
2
(
x
) =
x,
f
3
(
x
) =
x
2
, ...,
f
n
+1
(
x
) =
x
n
are linearly independent on the real line. Use this result to prove that, for
any constant
r
f
1
(
x
) =
e
rx
,
f
2
(
x
) =
xe
rx
,
f
3
(
x
) =
x
2
e
rx
, ...,
f
n
+1
(
x
) =
x
n
e
rx
are linearly independent on the real line.
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 Spring '08
 SivabalSivaloganathan
 real line, linear homogeneous ODE

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