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 roots ( 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 = c 1 e a x cos β x + c 2 e a x sin β x = e a x ( c 1 cos β x + c 2 sin β x ) Calculation: Consider the given differential equation y ″ − y ′ − 6 y = 0

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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The general solution of the second order differential equation y ″ − y ′ − 6 y = 0 .

Solution Summary: The author explains the general solution of the second order differential equation, y′′-xprime, where a,bandc are constants.

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The general solution of the second order differential equation y″−y′−6y=0.

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