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 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 the particular solution y p ( t ) . 6. Find the fundamental set as, If S = { y 1 ( t ) , y 2 ( t ) } Then, the wronskian is, W ( s ) = | y 1 ( t ) y 2 ( t ) d d t ( y 1 ( t ) ) d d t ( y 2 ( t ) ) | 7. Calculate u 1 ′ u 2 ′ and u 3 ′ as, u k ′ = { a i j | a i j ∈ W ( s ) except a i k = − c } 8. Integrate u 1 ′ u 2 ′ and u 3 ′ to get the values of u 1 , u 2 and u 3 . 9. The particular solution is given as y p ( t ) = u 1 y 1 + u 2 y 2 + u 3 y 3 10. The solution of the differential equation is given as y ( t ) = y h ( t ) + y p ( t ) . Calculation: Consider the differential equation. y ‴ − 4 y ″ − 11 y ′ + 30 y = e 4 t ⋯ ⋯ ( 1 ) The characteristic equation is, r 3 − 4 r 2 − 11 r + 30 = 0 The roots of the above equation are, r 3 − 4 r 2 − 11 r + 30 = 0 ( r − 5 ) ( r + 3 ) ( r − 2 ) = 0 Here, r = − 3 , k = 1 , r = 5 , k = 1 and r = 2 , k = 1

5th Edition

ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Question

Chapter 4.6, Problem 20E

To determine

The solution of the differential equation y‴−4y″−11y′+30y=e4t by any method.

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Chapter 4.6, Problem 19E

Chapter 4.6, Problem 21E

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Chapter 4 Solutions

Introductory Differential Equations

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The solution of the differential equation y ‴ − 4 y ″ − 11 y ′ + 30 y = e 4 t by any method.

The solution of the differential equation y‴−4y″−11y′+30y=e4t by any method.

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Chapter 4 Solutions