To determine The solution of the given IVP. Explanation Given: The given IVP is ( D 3 + 3 D 2 + 3 D + I ) y = 8 sin x , y ( 0 ) = − 1 , y ′ ( 0 ) = − 3 , y ″ ( 0 ) = 5 . Procedure used: Let y ( n ) + a n − 1 ( x ) y ( n − 1 ) + ⋯ + a 1 ( x ) y ′ + a 0 ( x ) y = r ( x ) be the n th order homogeneous ODE with constant coefficients, where y ( n ) = d n y d x n is the first term and r ( x ) ≠ 0 . A general solution of the above equation on an open interval I of the x -axis is of the form r ( x ) ≠ 0 Obtain the characteristic equation by substituting y = e λ x in the ODE as follows. y ( x ) = y h ( x ) + y p ( x ) Here y h ( x ) is a general solution of the corresponding homogeneous ODE. And y p ( x ) is any solution of the given equation on I containing no arbitrary constants. Calculation: Consider the given IVP, ( D 3 + 3 D 2 + 3 D + I ) y = 8 sin x , y ( 0 ) = − 1 , y ′ ( 0 ) = − 3 , y ″ ( 0 ) = 5 y ′ ′ ′ + 3 y ″ + 3 y ′ + y = 8 sin x , y ( 0 ) = − 1 , y ′ ( 0 ) = − 3 , y ″ ( 0 ) = 5 Consider the corresponding homogeneous equation of the given ODE, y ′ ′ ′ + 3 y ″ + 3 y ′ + y = 0 Obtain the characteristic equation as follows. λ 3 + 3 λ 2 + 3 λ + 1 = 0 Compute the roots of the characteristic equation as follows. λ 3 + 3 λ 2 + 3 λ + 1 = 0 ( λ + 1 ) ( λ 2 + 2 λ + 1 ) = 0 ( λ + 1 ) 3 = 0 λ 1 = λ 2 = λ 3 = − 1 The homogeneous solution of the given ODE is given as follows. y h = c 1 e − x + c 2 x e − x + c 3 x 2 e − x Let the particular solution is of the form y p = A cos x + B sin x . Obtain the first three derivatives of the above equation as follows. y ′ p = − A sin x + B cos x y ″ p = − A cos x − B sin x y ′ ′ ′ p = A sin x − B cos x Substitute the above derivatives in the given ODE as follows. A sin x − B cos x + 3 ( − A cos x − B sin x ) + 3 ( − A sin x + B cos x ) + ( A cos x + B sin x ) = 8 sin x cos x ( − B − 3 A + 3 B + A ) + sin x ( A − 3 B − 3 A + B ) = 8 sin x cos x ( 2 B − 2 A ) + sin x ( − 2 B − 2 A ) = 8 sin x Equate like terms on both sides as follows. 2 B − 2 A = 0 and − 2 B − 2 A = 8 A = B and − 2 A − 2 A = 8 A = B and − 4 A = 8 A = B = − 2 The particular solution of the given ODE is given as follows

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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Chapter 3, Problem 20RQ

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