Explanation Procedure used: Solution to the differential equation with undetermined Coefficients-Annihilator Approach: Consider the differential equation L ( y ) = g ( x ) , Where the function g ( x ) may consists the finite sum, product of multiple polynomials, sine and cosine functions and exponential functions as well. 1 Find the complementary solution y c for the homogeneous equation L ( y ) = 0 . 2 Operate on both sides of the equation L ( y ) = g ( x ) by the differential operator that will annihilate the function g ( x ) . 3 Find the general solution to the differential equation obtained L 1 L ( y ) = 0 . This is a homogeneous differential equation. 4 Remove the duplicated term in the complementary solution of L 1 L ( y ) = 0 . This gives the particular solution to the nonhomogeneous differential equation L ( y ) = g ( x ) . Calculation: Consider the differential equation y ″ − y = 1 + e x . The associated homogeneous equation is y ″ − y = 0 . First solve the homogeneous equation y ″ − y = 0 . The auxiliary equation for the homogenous differential equation y ″ − y = 0 is m 2 − 1 = 0 . Therefore, m 2 − 1 = 0 m 2 = 1 m = ± 1 Thus, solutions to the auxiliary equation are m 1 = 1 and m 2 = − 1 . Note that, both the roots are real and distinct Thus, the complementary solution to the homogeneous differential equation y ″ − y = 0 is, y c = c 1 e x + c 2 e − x . Since, the differential operator D annihilates 1 and ( D − 1 ) annihilates e x

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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The blank in the statement, “For the method of undermined coefficients, the assumed form of the particular solution for y ″ − y = 1 + e x is _____.”

Solution Summary: The author explains how to solve the homogeneous differential equation with undetermined Coefficients-Annihilator Approach.

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