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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 c,,C¡p(n)+c2q(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 = - r2; (ii) a, = nr" is also a root of (*).

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 c,,C¡p(n)+c2q(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 = - r2; (ii) a, = nr" is also a root of (*).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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DO PART (B) Show steps clearly with answer circled. No cursive if possible. 

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 c,,C¡p(n)+c2q(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 = - r2; (ii) a, = nr" is also a root of (*).
Transcribed Image Text: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 c,,C¡p(n)+c2q(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 = - r2; (ii) a, = nr" is also a root of (*).
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