To determine To show: The second solution y 2 satisfies ( y 2 y 1 ) ′ = W ( y 1 , y 2 ) y 1 2 and determine the second solution y 2 . Explanation The given differential equation is y ″ + p ( t ) y ′ + q ( t ) y = 0 . Let y 1 and y 2 are both solutions to the differential equation. The quotient rule of derivatives is computed as follows. ( y 2 y 1 ) ′ = y 1 ( t ) y ′ 2 ( t ) − y 2 ( t ) y ′ 1 ( t ) y 1 2 ( t ) = W ( y 1 , y 2 ) y 1 2 ( t ) Here, W ( y 1 , y 2 ) is the Wronskian of y 1 and y 2 . Hence proved. Note that, the Abel’s formula is W ( y 1 , y 2 ) ( t ) = c e − ∫ p ( t ) d t . Substitute W ( y 1 , y 2 ) ( t ) = c e − ∫ p ( t ) d t in ( y 2 y 1 ) ′ = W ( y 1 , y 2 ) y 1 2 ( t )

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 3.4, Problem 32P

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To show: The second solution y2 satisfies (y2y1)′=W(y1,y2)y12 and determine the second solution y2.

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Chapter 3.4, Problem 31P

Chapter 3.4, Problem 33P

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Elementary Differential Equations

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