This preview shows pages 1–2. Sign up to view the full content.
MATH11007 SHEET 15: DIFFERENCE EQUATIONS
Set on Tuesday, April 21: Qs 1, 2 and 3.
(1) Solve the following diﬀerence equations:
(a)
x
n
+2

x
n
= 0
(b)
x
n
+2
+ 3
x
n
+1

7
4
x
n
= 9
(c)
x
n
+2

3
x
n
+1

4
x
n
= 36
n
, with
x
1
= 1 and
x
2
= 5
(d)
x
n
+2
+ 2
x
n
+1
+
x
n
= 9
×
2
n
(2) Consider a diﬀerence equation
x
n
+2
+
ax
n
+1
+
b
= 0. Write down a condition
on
a
and
b
such that the characteristic equation,
m
2
+
am
+
b
= 0, has a
double root (i.e. only one solution). If ˜
m
is the unique solution of the
characteristic equation, show that
x
n
=
n
˜
m
n
is a solution of the diﬀerence
equation.
(3) For
n
∈
N
, deﬁne
T
n
(
x
) := cos (
n
arccos
x
)
.
(a) Show that
T
0
and
T
1
are polynomials of degree 0 and 1 respectively.
(b) Show that
T
n
+1
(
x
) +
T
n

1
(
x
) = 2
xT
n
(
x
)
.
Hence, prove by induction on
n
that
T
n
is a polynomial of degree
n
.
These polynomials are called the
Chebyshev
polynomials.
(c) Find the roots of
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.
 Fall '06
 Richards
 Equations

Click to edit the document details