To determine The particular solution of the given differential equation by using the method of variation of parameters and check the result by using the method of undetermined coefficient. Explanation The given differential equation is y ″ + 5 y ′ + 6 y = 2 e t . “Note that, the general solution of the differential equation y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y = g ( t ) is in the form of Y ( t ) = u 1 ( t ) y 1 ( t ) + u 2 ( t ) y 2 ( t ) , where y 1 ( t ) and y 2 ( t ) form a fundamental set of solution of the homogenous differential equation y ″ ( t ) + p ( t ) y ′ ( t ) + q ( t ) y = 0 , here u 1 ( t ) = − ∫ y 2 ( t ) g ( t ) W [ y 1 , y 2 ] ( t ) d t and u 2 ( t ) = ∫ y 1 ( t ) g ( t ) W [ y 1 , y 2 ] ( t ) d t .” Obtain the complementary solution of the given differential equation. The characteristic equation is r 2 + 5 r + 6 = 0 . r 2 − 5 r + 6 = 0 r 2 − 2 r − 3 r + 6 = 0 r ( r − 2 ) − 3 ( r − 2 ) = 0 ( r − 3 ) ( r − 2 ) = 0 On further simplification, r − 2 = 0 or r − 3 = 0 r = 2 or r = 3 The characteristic roots are r 1 = 2 or r 2 = 3 . Thus, the complementary solution is y c ( t ) = c 1 e 2 t + c 2 e 3 t . Note that, the functions y 1 = e 2 t and y 2 = e 3 t are the fundamental set of solutions of the homogeneous equation. Find the wronskian of the functions y 1 = e 2 t and y 2 = e 3 t . W [ y 1 , y 2 ] = | y 1 y 2 y ′ 1 y ′ 2 | = | e 2 t e 3 t 2 e 2 t 3 e 3 t | = 3 e 3 t ⋅ e 2 t − 2 e 2 t ⋅ e 3 t = 3 e 5 t − 2 e 5 t = e 5 t Obtain the value of u 1 ( t ) = − ∫ y 2 ( t ) g ( t ) W [ y 1 , y 2 ] ( t ) d t . u 1 ( t ) = − ∫ y 2 ( t ) g ( t ) W [ y 1 , y 2 ] ( t ) d t = − ∫ e 3 t ( 2 e t ) e 5 t d t = − ∫ 2 e 4 t e 5 t d t = − 2 ∫ 1 e t d t On further simplification, u 1 ( t ) = − 2 ∫ e − t d t = − 2 e − t − 1 = 2 e − t Obtain the value of u 2 ( t ) = − ∫ y 1 ( t ) g ( t ) W [ y 1 , y 2 ] ( t ) d t

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 3.6, Problem 1P

To determine

The particular solution of the given differential equation by using the method of variation of parameters and check the result by using the method of undetermined coefficient.

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Chapter 3.5, Problem 39P

Chapter 3.6, Problem 2P

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Elementary Differential Equations

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