Explanation Given: The given condition is: c 1 u 1 + ⋯ + c k u k + c k + 1 u k + 1 = 0 and the induction hypothesis that u 1 , ⋯ , u k , 2 ≤ k are linear independent. Proof: Let A be a matrix of order k + 1 by k + 1 . Let us suppose A has k + 1 distinct eigenvalues r 1 , r 2 , ⋯ , r k and r k + 1 . Let the eigenvectors with respect to these eigenvalues be denoted by u 1 , u 2 , ⋯ , u k and u k + 1 . Let, c 1 u 1 + ⋯ + c k u k + c k + 1 u k + 1 = 0 Multiply both sides by the matrix A. we get: A ( c 1 u 1 + ⋯ + c k u k + c k + 1 u k + 1 ) = A ( 0 ) c 1 Au 1 + ⋯ + c k Au k + c k + 1 Au k + 1 = 0 c 1 r 1 u 1 + ⋯ + c k r k u k + c To determine (b) To prove: The expressions u 1 , ⋯ , u k + 1 are linear independent for the given condition.

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 9.5, Problem 48E

To determine

(a)

To prove:

The expression c1(r1−rk+1)u1+⋯+ck(rk−rk+1)uk=0 for the given condition.

To determine

(b)

To prove:

The expressions u1,⋯,uk+1 are linear independent for the given condition.

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Chapter 9.5, Problem 46E

Chapter 9.5, Problem 49E

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Chapter 9 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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(a) To prove: The expression c 1 ( r 1 − r k + 1 ) u 1 + ⋯ + c k ( r k − r k + 1 ) u k = 0 for the given condition.

Solution Summary: The author explains the induction hypothesis that u_1,cdots

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(a)

To prove:

The expression c1(r1−rk+1)u1+⋯+ck(rk−rk+1)uk=0 for the given condition.

(b)

To prove:

The expressions u1,⋯,uk+1 are linear independent for the given condition.

Want to see the full answer?

Chapter 9 Solutions

(a) To prove: The expression c 1 ( r 1 − r k + 1 ) u 1 + ⋯ + c k ( r k − r k + 1 ) u k = 0 for the given condition.

Solution Summary: The author explains the induction hypothesis that u_1,cdots

Concept explainers

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(a) To prove: The expression c1(r1−rk+1)u1+⋯+ck(rk−rk+1)uk=0 for the given condition.

(b) To prove: The expressions u1,⋯,uk+1 are linear independent for the given condition.

Want to see the full answer?

Chapter 9 Solutions

(a)

To prove:

The expression c1(r1−rk+1)u1+⋯+ck(rk−rk+1)uk=0 for the given condition.

(b)

To prove:

The expressions u1,⋯,uk+1 are linear independent for the given condition.