Explanation Consider boundary value problem as follows: x 2 y ″ + x y ' + 9 λ y = 0 ...... ( 1 ) Assume x = e z and simplify the substitution as follows: x = e z log x = z ( Take logarithm both sides ) 1 x d x d y = d z d y ( differentiate both sides with respect to y ) d y d z = x d y d x Further, simplify the above equation as follows: d 2 y d z 2 = d d z ( x d y d x ) ( differentiate both sides with respect to z ) d 2 y d z 2 = x d d z ( d y d x ) + d y d x d d z ( x ) d 2 y d z 2 = x d d z ( d y d x ) + d y d x ( x ) ( d x d z = x ) d 2 y d z 2 = x ( x d 2 y d x 2 ) + d y d x ( x ) ( d d z ( d y d x ) = x d 2 y d x 2 ) Further, simplify the above equation as follows: d 2 y d z 2 = x 2 d 2 y d x 2 + x d y d x d 2 y d z 2 = x 2 y ″ + x y ′ So, equation (1) can be written as follows: d 2 y d z 2 + 9 λ y = 0 ( d 2 y d z 2 = x 2 y ″ + x y ′ ) ...... ( 2 ) Write the Auxiliary equation of equation (2) as follows: m 2 + 9 λ = 0 m 2 = − 9 λ When λ = 0 or λ is negative, then the obtain solution is trivial as below: y ″ = 0 , λ = 0 ...... ( 3 ) The general solution of the differential equation (2) is as follows: y ( z ) = c 1 + c 2 z

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 11, Problem 19RE

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The Eigen value and Eigen function of boundary value problem x2y″+xy'+9λy=0, y'(1)=0,y(e)=0.

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Chapter 11, Problem 18RE

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

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The Eigen value and Eigen function of boundary value problem x 2 y ″ + x y ' + 9 λ y = 0 , y ' ( 1 ) = 0 , y ( e ) = 0 .

Solution Summary: The author explains the Eigen value and Eigen function of boundary value problem x2y′′+xy

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The Eigen value and Eigen function of boundary value problem x2y″+xy'+9λy=0, y'(1)=0,y(e)=0.

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