To determine The general solution of the given nonhomogeneous linear ordinary differential equation ( D 2 + 2 D + I ) y = 2 x sin x . Explanation Result used: Basic rule : In this rule, determine the undetermined coefficients of y p by substituting y p and its derivatives in y ″ + a y ′ + b y = r ( x ) . Calculation: The given ordinary differential equation is y ″ + 2 y ′ + y = 2 x sin x . Obtain the solution y h of the homogeneous differential equation y ″ + 2 y ′ + y = 0 as follows. The auxiliary equation of y ″ + 2 y ′ + y = 0 is m 2 + 2 m + 1 = 0 . Now obtain the roots of the auxiliary equation m 2 + 2 m + 1 = 0 as shown below. m 2 + 2 m + 1 = 0 m 2 + m + m + 1 = 0 ( m + 1 ) 2 = 0 m = − 1 ( double root ) Here the root is a double real root. Therefore the solution y h is y h = ( c 1 + c 2 x ) e − x . Now obtain the particular solution y p as follows. The choice for y p is y p = K x cos x + M x sin x + N cos x + P sin x . Differentiate y p with respect to x and obtain the first derivative as shown below. y ′ p 1 = d d x ( K x cos x + M x sin x + N cos x + P sin x ) = K ( − x sin x + cos x ) + M ( x cos x + sin x ) − N sin x + P cos x = − K x sin x + K cos x + M x cos x + M sin x − N sin x + P cos x = ( − K x + M − N ) sin x + ( M x + K + P ) cos x Again differentiate y ′ p 1 and obtain the second derivative as follows. y ″ p 1 = d d x ( ( − K x + M − N ) sin x + ( M x + K + P ) cos x ) = ( − K x + M − N ) cos x − K sin x − ( M x + K + P ) sin x + M cos x = ( − K x + 2 M − N ) cos x − ( M x + 2 K + P ) sin x Substitute the obtained derivatives in y ″ + 2 y ′ + y = 2 x sin x and determine the undetermined coefficients as shown below. { ( − K x + 2 M − N ) cos x − ( M x + 2 K + P ) sin x + 2 ( ( − K x + M − N ) sin x + ( M x + K + P ) cos x ) + K x cos x + M x sin x + N cos x + P sin x } = 2 x sin x { ( 2 M + 2 K + 2 P ) cos x + ( 2 M − 2 K − 2 N ) sin x − 2 K x sin x + 2 M x cos x } = 2 x sin x Equate the coefficients of like terms on both sides as follows

10th Edition

ISBN: 9780470458365

Author: Erwin Kreyszig

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2.7, Problem 10P

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The general solution of the given nonhomogeneous linear ordinary differential equation (D2+2D+I)y=2xsinx.

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