Unformatted text preview: MTHSC 208 (Diﬀerential Equations)
Dr. Matthew Macauley
HW 16
Due Monday March 23rd, 2009
(1) Use the ratio test to ﬁnd the radius of convergence of the following power series:
∞ 1
(x − π )n ,
n+1
n=0 b. n=0
∞ d. ∞ ∞ (−1)n xn , a. 1
(x − π )n ,
2n
n=0 c. ∞ ∞ (5x − 10)n , e. 3
(x − 2)n ,
n+1
n=0 f. n=0 1
(3x − 6)n .
n!
n=0 (2) Use the comparison test to ﬁnd an estimate for the radius of convergence of each of the
following power series:
∞ a. ∞ 1
x2n ,
(2n)!
n=0 (−1)n x2n , b. n=0
∞ ∞ c. 1
(x − 4)2n ,
2n
n=1 1
(x − π )2n .
22 n
n=0 d. (3) Find the radius of convergence of the power series
∞ (−1)n
n=0 1
(n!)2 x
2 2n . (4) The diﬀerential equation
dy
d2 y
−x
+ p2 y = 0,
2
dx
dx
where p is a constant, is known as Chebyshev’s equation. It can be rewritten in the form
(1 − x2 ) d2 y
dy
x
p2
− P (x)
+ Q(x)y = 0,
where
P (X ) = −
, Q(x) =
.
2
2
dx
dx
1−x
1 − x2
(a) If P (x) and Q(x) are represented as a power series about x0 = 0, what is the radius
of convergence of these power series?
(b) Assuming a power series centered at 0, ﬁnd the recursion formula for an+2 in terms
of an .
(c) Use the recursion formula to determine an in terms of a0 and a1 , for 2 ≤ n ≤ 9.
(d) In the special case where p = 3, ﬁnd a nonzero polynomial solution to this diﬀerential
equation. ...
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This note was uploaded on 03/12/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.
 Spring '09
 Staufeneger
 Differential Equations, Equations, Power Series

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