Explanation Theorem used: 1. The set H of all solutions of the n th -order homogeneous linear difference equation y k + n + a 1 y k + n + 1 + ... + a n − 1 y k + 1 + a n y k = 0 for all k is an n -dimensional vector space. 2. A linearly independent set of n vectors in an n -dimensional space is automatically a basis and span the vector space. Calculation: To verify the signal y k = 2 − 2 k is a solution of the difference equation y k + 2 − 9 2 y k + 1 + 2 y k = 2 + 3 k . It is given that y k = 2 − 2 k . Thus, y k + 1 = 2 − 2 ( k + 1 ) = − 2 k and y k + 2 = 2 − 2 ( k + 2 ) = − 2 − 2 k . Substitute the value y k = 2 − 2 k , y k + 1 = − 2 k and y k + 2 = − 2 − 2 k in y k + 2 − 8 y k + 1 + 15 y k = 2 + 8 k . y k + 2 − 9 2 y k + 1 + 2 y k = ( − 2 − 2 k ) − 9 2 ( − 2 k ) + 2 ( 2 − 2 k ) = − 2 − 2 k + 9 k + 4 − 4 k = 3 k + 2 Therefore, the signal y k = 2 − 2 k is a solution of the difference equation y k + 2 − 9 2 y k + 1 + 2 y k = 2 + 3 k . The solution of the homogenous difference equation is of the form y k = r k for some r . Substitute r k for y k in the difference equation y k + 2 − 9 2 y k + 1 + 2 y k = 0 and simplify the auxiliary equation r 2 − 9 2 r + 2 = 0 as follows

Linear Algebra and Its Applications (5th Edition)

5th Edition

ISBN: 9780321982384

Author: David C. Lay, Steven R. Lay, Judi J. McDonald

Publisher: PEARSON

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Chapter 4.8, Problem 27E

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To show that: The signal yk=2−2k is a solution of the difference equation yk+2−92yk+1+2yk=2+3k and find the general solution of yk+2−92yk+1+2yk=2+3k.

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Chapter 4.8, Problem 26E

Chapter 4.8, Problem 28E

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Linear Algebra and Its Applications (5th Edition)

Ch. 4.1 - Show that the set H of all points in 2 of the form...Ch. 4.1 - Let W = Span{v1,...,vp}, where v1,..,vp are in a...Ch. 4.1 - An n n matrix A is said to be symmetric if AT =...Ch. 4.1 - Let V be the first quadrant in the xy-plane; that...Ch. 4.1 - Let W be the union of the first and third...Ch. 4.1 - Let H be the set of points inside and on the unit...Ch. 4.1 - Construct a geometric figure that illustrates why...Ch. 4.1 - In Exercises 58, determine if the given set is a...Ch. 4.1 - In Exercises 58, determine if the given set is a...Ch. 4.1 - In Exercises 58, determine if the given set is a...

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To show that: The signal y k = 2 − 2 k is a solution of the difference equation y k + 2 − 9 2 y k + 1 + 2 y k = 2 + 3 k and find the general solution of y k + 2 − 9 2 y k + 1 + 2 y k = 2 + 3 k .

Solution Summary: The author explains that the signal y_k=2-2k is a solution of the difference equation.

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To show that: The signal yk=2−2k is a solution of the difference equation yk+2−92yk+1+2yk=2+3k and find the general solution of yk+2−92yk+1+2yk=2+3k.

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Chapter 4 Solutions