Explanation Result used: (1) “Suppose the equation L [ x r ] = x 2 y ″ + α x y ′ + β y = 0 has a solution of the from y = x r , then it becomes L [ x r ] = x r [ r ( r − 1 ) + α r + β ] , where α and β are real constants”. (2) “If r is a root of the quadratic equation F ( r ) = r ( r − 1 ) + α r + β = 0 , then L [ x r ] is zero and y = x r is a solution of x 2 y ″ + α x y ′ + β y = 0 and the roots of the equation are r 1 , r 2 = − ( α − 1 ) ± ( α − 1 ) 2 − 4 β 2 ”. (3) If the roots r 1 = λ + i μ and r 2 = λ − i μ of the equation are complex, then the general solution is in the form of y = c 1 x λ cos ( μ ln x ) + c 2 x λ sin ( μ ln x ) , x > 0 ”. Calculation: The given differential equation is ( x − 2 ) 2 y ″ + 5 ( x − 2 ) y ′ + 8 y = 0 . Assume that the solution is in the form of y = ( x − 2 ) r . Substitute y = ( x − 1 ) r in the equation ( x − 2 ) 2 y ″ + 5 ( x − 2 ) y ′ + 8 y = 0 as follows

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Question

Chapter 5.4, Problem 8P

To determine

The general solution of the differential equation (x−2)2y″+5(x−2)y′+8y=0.

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Chapter 5.4, Problem 7P

Chapter 5.4, Problem 9P

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Elementary Differential Equations and Boundary Value Problems, Enhanced

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