Explanation Procedure used: 1. Make the characteristic equation by substituting y ″ = r 2 , y ′ = r . The characteristic equation is, a r 2 + b r + c = 0 Here, the coefficient of y ″ is a , the coefficient of y ′ is b and the coefficient of y is c . 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 5. Find particular solution y p ( t ) . 6. The solution of differential equation is given as y ( t ) = y h ( t ) + y p ( t ) . Calculation: Consider the differential equation. y ‴ − 13 y ′ + 12 y = cos t (1) The characteristic equation is, r 3 − 13 r + 12 = 0 The roots of the above equation are, r 3 − 13 r + 12 = 0 ( r + 4 ) ( r − 3 ) ( r − 1 ) = 0 Hence, r = 3 , k = 1 , r = 1 , k = 1 and r = − 4 , k = 1 . For real root, the fundamental set with respect to r = 3 , k = 1 . S 1 = e 3 × t S 1 = e 3 t For real root, the fundamental set with respect to r = 1 , k = 1 . S 2 = e 1 × t S 2 = e t For real root, the fundamental set with respect to r = − 4 , k = 1 . S 3 = e − 4 × t S 3 = e − 4 t The fundamental set of solution is, S = { e 3 t , e t , e − 4 t } The general solution is, y h ( t ) = c 1 e 3 t + c 2 e t + c 3 e − 4 t (2) Consider the forcing function

5th Edition

ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Chapter 4.6, Problem 22E

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The solution of y‴−13y′+12y=cost.

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Chapter 4.6, Problem 21E

Chapter 4.6, Problem 23E

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Topic: Group Theory | Abstract Algebra Question: Let G be a finite group of order 45. Prove that G has a normal subgroup of order 5 or order 9, and describe the number of Sylow subgroups for each. Instructions: • Use Sylow's Theorems (existence, conjugacy, and counting). • List divisors of 45 and compute possibilities for n for p = 3 and p = 5. Show that if n = 1, the subgroup is normal. Conclude about group structure using your analysis.

Chapter 4 Solutions

Introductory Differential Equations

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The solution of y ‴ − 13 y ′ + 12 y = cos t .

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Chapter 4 Solutions