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 quadratic form of the given differential equation is r ( r − 1 ) + α r + β = 0 . Solve the equation r ( r − 1 ) + α r + β = 0 by quadratic formula. r 1 , r 2 = 1 − α ± ( α − 1 ) 2 − 4 ( 1 ) ( β ) 2 ( 1 ) = 1 − α ± ( α − 1 ) 2 − 4 β 2 If the equation has repeated roots, then ( α − 1 ) 2 − 4 β = 0 . That is, r 1 , 2 = 1 − α 2 ...... ( 1 ) . Let r 1 be the root of the equation. Then r 1 ( r 1 − 1 ) + α r 1 + β = 0 ...... ( 2 ) . Assume that one solution of the initial differential equation is y 1 ( x ) = x r 1 . By the method of reduction of order the solution y is y ( x ) = v ( x ) x r 1 . Find the first derivative of y ( x ) = v ( x ) x r 1 . y ′ ( x ) = v ′ ( x ) x r 1 + r 1 v ( x ) x r 1 − 1 The second derivative is computed as follows. y ″ ( x ) = v ″ ( x ) x r 1 + r 1 v ′ ( x ) x r 1 − 1 + r 1 v ′ ( x ) x r 1 − 1 + r 1 ( r 1 − 1 ) v ( x ) x r 1 − 2 = v ″ ( x ) x r 1 + 2 1 v ′ ( x ) x r 1 − 1 + r 1 ( r 1 − 1 ) v ( x ) x r 1 − 2 Substitute y ″ ( x ) and y ′ ( x ) in the equation x 2 y ″ + α x y ′ + β y = 0

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 28P

To determine

To show: If r1 is repeated root of r(r−1)+αr+β=0, then xr1 and xr1lnx are the solutions of x2y″+αxy′+βy=0 for x>0.

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Chapter 5.4, Problem 27P

Chapter 5.4, Problem 29P

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Chapter 5 Solutions

Elementary Differential Equations and Boundary Value Problems, Enhanced

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