Explanation Procedure used: Solution to a second order differential equation. For a second order differential equation a y ″ + b y ′ + c y = 0 , where a , b and c are constants, the auxiliary equation will be of the form a m 2 + b m + c = 0 . There will be three forms of the general solution of a y ″ + b y ′ + c y = 0 corresponding to three cases which dependent to the solution of auxiliary equation. 1. m 1 and m 2 are real and distinct roots that is possible when ( b 2 − 4 a c > 0 ) 2. m 1 and m 2 are real and equal roots that is possible when ( b 2 − 4 a c = 0 ) 3. m 1 and m 2 are conjugate complex root ( b 2 − 4 a c < 0 ) Case I: Distinct Real Roots The general solution of a y ″ + b y ′ + c y = 0 on the interval ( − ∞ , ∞ ) will be y = c 1 e m 1 x + c 2 e m 2 x . Case II: Repeated Real Roots The general solution will be y = c 1 e m 1 x + c 2 x e m 1 x . Case III: Conjugate Complex Roots If m 1 and m 2 are complex roots, then m 1 = α + i β and m 2 = α − i β , where α and β > 0 are real and i 2 = − 1 . The general solution will be y = e a x ( c 1 cos β x + c 2 sin β x ) . Calculation: Consider the given differential equation 3 y ″ + y = 0 . Consider y = e m x as the solution of the differential equation 3 y ″ + y = 0 . Substitute y = e m x , and y ″ = m 2 e m x into 3 y ″ + y = 0 to obtain the auxiliary equation. 3 m 2 e m x + e m x = 0 e m x ( 3 m 2 + 1 ) = 0 Solve the auxiliary equation as follows

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.3, Problem 10E

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The general solution of the second order differential equation 3y″+y=0.

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The general solution of the second order differential equation 3 y ″ + y = 0 .

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