To determine The solution of the initial value problem ( D 2 + 2 D + 10 I ) y = 17 sin x − 37 sin 3 x , y ( 0 ) = 6.6 , y ′ ( 0 ) = − 2.2 . Explanation Formula used: Method of undetermined coefficients: Consider the linear ordinary differential equation y ″ + a y ′ + b y = r ( x ) . 1. If the term in r ( x ) is of the form k e γ x , then the choice for y p ( x ) is C e γ x . 2. If the term in r ( x ) is of the form k x n ( n = 0 , 1 , … ) , then the choice for y p ( x ) is K n x n + K n − 1 x n − 1 + ⋯ + K 1 x + K 0 . 3. If the term in r ( x ) is of the form k cos ω x or k sin ω x , then the choice for y p ( x ) is K cos ω x + M sin ω x . 4. If the term in r ( x ) is of the form k e α x cos ω x or k e α x sin ω x , then the choice for y p ( x ) is e α x ( K cos ω x + M sin ω x ) . Calculation: The given initial value problem is ( D 2 + 2 D + 10 I ) y = 17 sin x − 37 sin 3 x , y ( 0 ) = 6.6 , y ′ ( 0 ) = − 2.2 . Obtain the solution y h of the homogeneous differential equation y ″ + 2 y ′ + 10 y = 0 as follows. The auxiliary equation of y ″ + 2 y ′ + 10 y = 0 is m 2 + 2 m + 10 = 0 . Now obtain the roots of the auxiliary equation m 2 + 2 m + 10 = 0 as shown below. m = − 2 ± 4 − 4 ( 10 ) 2 = − 2 ± − 36 2 = − 2 ± 6 i 2 = − 1 ± 3 i Here the roots are complex conjugate. Therefore the solution y h is y h = e − x ( c 1 cos 3 x + c 2 sin 3 x ) . Now obtain the particular solution y p as follows. By sum rule, y p = y p 1 + y p 2 The choice for y p 1 is y p 1 = K cos x + M sin x . Differentiate y p 1 with respect to x and obtain the first derivative as shown below. y ′ p 1 = d d x ( K cos x + M sin x ) = − K sin x + M cos x Again differentiate y ′ p 1 and obtain the second derivative as follows. y ″ p 1 = d d x ( − K sin x + M cos x ) = − K cos x − M sin x Substitute the obtained derivatives in y ″ + 2 y ′ + 10 y = 17 sin x and determine the undetermined coefficient as shown below. { ( − K cos x − M sin x ) + 2 ( − K sin x + M cos x ) + 10 ( K cos x + M sin x ) } = 17 sin x 9 K cos x + 9 M sin x − 2 K sin x + 2 M cos x = 17 sin x ( 9 K + 2 M ) cos x + ( 9 M − 2 K ) sin x = 17 sin x Equate the coefficients of like terms on both sides as follows. 9 K + 2 M = 0 … … ( 1 ) 9 M − 2 K = 17 … … ( 2 ) From equation (1), solve for K as follows. 9 K + 2 M = 0 9 K = − 2 M K = − 2 M 9 From equation (2), the value of M is obtained as shown below. 9 M − 2 K = 17 9 M − 2 ( − 2 M 9 ) = 17 81 M + 4 M = 17 ( 9 ) 85 M = 153 M = 1.8 Compute the value of K as follows. K = − 2 M 9 = − 2 ( 1.8 ) 9 = − 2 ( 0.2 ) = − 0.4 Therefore, the particular solution y p 1 is y p 1 = K cos x + M sin x = − 0.4 cos x + 1.8 sin x . Now the choice for y p 2 becomes K cos 3 x + M sin 3 x . Differentiate y p 2 with respect to x and obtain the first derivative as shown below. y ′ p 2 = d d x ( K cos 3 x + M sin 3 x ) = − 3 K sin 3 x + 3 M cos 3 x Again differentiate y ′ p 2 and obtain the second derivative as follows. y ″ p 2 = d d x ( − 3 K sin 3 x + M cos 3 x ) = − 3 K ( 3 cos 3 x ) − 9 M sin 3 x = − 9 K cos 3 x − 9 M sin 3 x Substitute the obtained derivatives in y ″ + 2 y ′ + 10 y = − 37 sin 3 x and determine the undetermined coefficients as shown below

10th Edition

ISBN: 9781119664697

Author: Kreyszig

Publisher: WILEY

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Chapter 2.7, Problem 18P

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The solution of the initial value problem (D2+2D+10I)y=17sinx−37sin3x, y(0)=6.6, y′(0)=−2.2.

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Chapter 2.7, Problem 17P

Chapter 2.7, Problem 19P

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