(a) Explanation Procedure used: 1. Make the characteristic equation by substituting y ′ = r , y ″ = r 2 , … . 2. Find the roots of the characteristic equation. 3. If r is real and k ≥ 2 , then The solution associated with r and k is, e r t , t e r t , t 2 e r t , ... , t k − 1 e r t 4. If r = α ± β i , then the solution associated are e α t cos β t . and e α t sin β t Calculation: Simplify the differential equation. y − k 4 d 4 y d x 4 = 0 k 4 d 4 y d x 4 − y = 0 d 4 y d x 4 − 1 k 4 y = 0 The characteristic equation is r 4 − 1 k 4 = 0 . The roots of the above equation is, r 4 − 1 k 4 = 0 ( r 2 − 1 k 2 ) ( r 2 + 1 k 2 ) = 0 Here, r = 1 k , k = 1 , r = − 1 k , k = 1 and r = ± 1 k i , k = 1 For real root the fundamental set with respect to r = − 1 k , k = 1 (b) To determine The conditions on the constants so that exactly one local maximum exists. (c) To determine The conditions on the constants so that exactly one local minimum exists.

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ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

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Question

Chapter 4.5, Problem 64E

(a)

To determine

The solution of y−k4d4ydx4=0.

(b)

To determine

The conditions on the constants so that exactly one local maximum exists.

(c)

To determine

The conditions on the constants so that exactly one local minimum exists.

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Chapter 4.5, Problem 62E

Chapter 4.5, Problem 65E

Chapter 4 Solutions

Introductory Differential Equations

Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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The solution of y − k 4 d 4 y d x 4 = 0 .

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The solution of y−k4d4ydx4=0.

The conditions on the constants so that exactly one local maximum exists.

The conditions on the constants so that exactly one local minimum exists.

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Chapter 4 Solutions