Explanation Given: The given boundary value problem is: x 2 y ″ ( x ) + 3 x y ′ ( x ) + 2 y ( x ) = 0 ; y ′ ( 1 ) = 0 , y ′ ( e π ) = 0 Procedure used: Let L be the differential operator given by the L [ y ] = A 2 y ″ + A 1 y ′ + A 0 y . The formal adjoint of L is the differential operator L + define by L + [ y ] = ( A 2 y ) ′ ′ − ( A 1 y ) ′ + A 0 y provided A 2 ′ ′ , A 1 ′ and A 0 are continuous on [ a , b ] . The Green’s formula: P ( u , v ) | a b = 0 , for u in D ( L ) , for v in D ( L + ) . Here P ( u , v ) , the bilinear concomitant associated with L , is define as P ( u , v ) = u A 1 v − u ( A 2 v ) ′ + u ′ A 2 v Let B [ u ] = 0 represent the boundary conditions required for a function to be in the domain of L and let B + [ u ] = 0 be the boundary conditions required for a function to be in the domain of its adjoint operator L + . Then, the adjoint boundary value problem for L [ u ] = 0 , B [ u ] = 0 is the boundary value problem L + [ u ] = 0 , B + [ u ] = 0 Calculation: Consider the given boundary value problem: x 2 y ″ ( x ) + 3 x y ′ ( x ) + 2 y ( x ) = 0 Here A 0 = 2 , A 1 = 3 x and A 2 = x 2 . Substitute in the formal adjoint formula: L + [ y ] = ( x 2 y ) ′ ′ − ( 3 x y ) ′ + 2 y = [ x 2 y ′ + 2 x y ] ′ − [ 3 x y ′ + 3 y ] + 2 y = [ x 2 y ″ + 2 x y ′ + 2 x y ′ + 2 y ] − 3 x y ′ − 3 y + 2 y = x 2 y ″ + x y ′ + y Apply the Green’s formula: P ( u , v ) ( x ) | 1 e π = 0 for all u in D ( L ) and v in D ( L + ) . Apply the formula of bilinear concomitant: ( ( 3 x ) u v − u ( x 2 v ) ′ + u ′ x 2 v ) | 1 e π = 0 ( 3 x u v − u ( x 2 v ′ + 2 x v ) + x 2 u ′ v ) | 1 e π = 0 ( { 3 e π u ( e π ) v ( e π ) − u ( e π ) ( e 2 π v ′ ( e π ) + 2 e π v ( e π ) ) + e 2 π u ′ ( e π ) v ( e π ) } − { 3 u ( 1 ) v ( 1 ) − u ( 1 ) ( v ′ ( 1 ) + 2 v ( 1 ) ) + u ′ ( 1 ) v ( 1 ) } ) = 0 For u in D ( L ) , we have u ′ ( 1 ) = 0 and u ′ ( e π ) = 0

7th Edition

ISBN: 9780321977175

Author: Nagle

Publisher: PEARSON CO

See similar textbooks

Videos

Question

Chapter 11.4, Problem 15E

To determine

To find:

The adjoint boundary value problem for the given boundary value problem x2y″(x)+3xy′(x)+2y(x)=0;y′(1)=0, y′(eπ)=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.4, Problem 14E

Chapter 11.4, Problem 16E

Students have asked these similar questions

How parents can assess children's learning at home and how the task can be differentiated. Must provide two examples of differentiation tasks. Mathematics in Practice Assignment 2

When ever one Point sets in X are closed a collection of functions which separates Points from closed set will separates Point. 18 (prod) is product topological space then xe A (xx, Tx) is homeomorphic to sub space of the Product space (TXA, prod). KeA The Bin Projection map 18: Tx XP is continuous and open but heed hot to be closed. Acale ctioneA} of continuos function ona topogical Space X se partes Points from closed sets inx iff the set (v) for KEA and Vopen set inx from a base for top on X-

Why are Bartleby experts giving only chatgpt answers?? Why are you wasting our Money and time ?

Chapter 11 Solutions

EBK FUND.OF DIFF.EQUATIONS+BOUNDARY...

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,...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The adjoint boundary value problem for the given boundary value problem x 2 y ″ ( x ) + 3 x y ′ ( x ) + 2 y ( x ) = 0 ; y ′ ( 1 ) = 0 , y ′ ( e π ) = 0 .

Solution Summary: The author explains that the adjoint boundary value problem is lx2y

Videos

To find:

The adjoint boundary value problem for the given boundary value problem x2y″(x)+3xy′(x)+2y(x)=0;y′(1)=0, y′(eπ)=0.

Want to see the full answer?

Chapter 11 Solutions

The adjoint boundary value problem for the given boundary value problem x 2 y ″ ( x ) + 3 x y ′ ( x ) + 2 y ( x ) = 0 ; y ′ ( 1 ) = 0 , y ′ ( e π ) = 0 .

Solution Summary: The author explains that the adjoint boundary value problem is lx2y

Videos

To find: The adjoint boundary value problem for the given boundary value problem x2y″(x)+3xy′(x)+2y(x)=0;y′(1)=0, y′(eπ)=0.

Want to see the full answer?

Chapter 11 Solutions

To find:

The adjoint boundary value problem for the given boundary value problem x2y″(x)+3xy′(x)+2y(x)=0;y′(1)=0, y′(eπ)=0.