Explanation Given: For Bessel’s equation of order v where 0 < v < 1 , the boundary condition (17) are replaced by lim x → 0 + y ( x ) = 0 , lim x → 0 + x y ′ ( x ) = 0 ( 33 ) if u and v satisfy the boundary conditions in (33), then equation (10) holds at a = 0 and hence the singular Sturm-Liouville boundary value problem (16), (18) (33) gives rise to a selfadjoint operator. The equation ( 10 ) as per text book is given by, lim x → a + p ( x ) [ u ( x ) − v ′ ( x ) − u ′ ( x ) v ( x ) ] = 0 . Approach: The singular Sturm-Liouville boundary value problem associated with Bessel’s equation of order v is given by, d d x [ x d y d x ] − v 2 x y + λ x y = 0 , 0 < x < 1 ... ( 1 ) y ( x ) , y ′ ( x ) remains bounded as x → 0 + and y ( 1 ) = 0 . The equation ( 10 ) as per text book is given by, lim x → a + p ( x ) [ u ( x ) v ′ ( x ) − u ′ ( x ) v ( x ) ] = 0 . The operator L will be a selfadjoint if the right hand side of the equation ( 10 ) is zero To determine (b) To show: That J v ( λ x ) satisfies the boundary conditions in the given equation ( 33 ) .

Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

7th Edition

ISBN: 9780134768717

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Chapter 11.7, Problem 10E

To determine

(a)

To show:

That the linear operator is selfadjoint.

To determine

(b)

To show:

That Jv(λx) satisfies the boundary conditions in the given equation (33).

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Chapter 11.7, Problem 9E

Chapter 11.7, Problem 11E

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Fundamentals Of Differential Equations And Boundary Value Problems Plus Mylab Math With Pearson Etext -- Title-specific Access Card Package (7th ... Fundamentals Of Differential Equations)

Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 3E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 7E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...Ch. 11.2 - Prob. 9E Ch. 11.2 - In Problems 1-12, determine the solutions, if any,...

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(a) To show: That the linear operator is selfadjoint.

Solution Summary: The author explains that the linear operator is selfadjoint in Bessel's equation of order v.