6.34. Consider the following second-order recurrence relation and its characteristic polynomial A(x): a, = sa,-1 + ta„-2 and A(x) = x² – sx – t (*) (a) Suppose p(n) and q(n) are solutions of (*). Show that, for any constants c, and c2, cp(n)+c,q(n) is also a solution of (*). (b) Suppose r is a root of A(x). Show that a, = r" is a solution to (*). (c) Suppose r is a double root of A(x). Show that: (i) s = : 2r and t =-i -r?; (ii) a, = nr" is also a root of (*).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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6.34. Consider the following second-order recurrence relation and its characteristic polynomial A(x):
a, = sa,-1 + ta„-2 and A(x) = x² – sx – t
(*)
(a) Suppose p(n) and q(n) are solutions of (*). Show that, for any constants c, and c2, cp(n)+c,q(n) is also a solution
of (*).
(b) Suppose r is a root of A(x). Show that a, = r" is a solution to (*).
(c) Suppose r is a double root of A(x). Show that: (i) s =
: 2r and t =-i
-r?; (ii) a, = nr" is also a root of (*).
Transcribed Image Text:6.34. Consider the following second-order recurrence relation and its characteristic polynomial A(x): a, = sa,-1 + ta„-2 and A(x) = x² – sx – t (*) (a) Suppose p(n) and q(n) are solutions of (*). Show that, for any constants c, and c2, cp(n)+c,q(n) is also a solution of (*). (b) Suppose r is a root of A(x). Show that a, = r" is a solution to (*). (c) Suppose r is a double root of A(x). Show that: (i) s = : 2r and t =-i -r?; (ii) a, = nr" is also a root of (*).
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