Baixe EQUAÇÕES DIFERENCIAIS - resoluções Boyce e DiPrima - equaes cap. xi e outras Notas de estudo em PDF para Engenharia Civil, somente na Docsity! —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 720 Chapter Eleven Section 11.1 1. Since the right hand sides of the ODE and the boundary conditions are all zero, the boundary value problem is .homogeneous 3. The right hand side of the ODE is . Therefore the boundary value problem isnonzero nonhomogeneous. 6. The ODE can also be written as C  "  B C œ !ww #-ˆ ‰ . Although the second boundary condition has a more general form, the boundary value problem is .homogeneous 7. First assume that . The general solution of the ODE is . The- œ ! C B œ - B  -a b " # boundary condition at requires that . Imposing the second condition,B œ ! - œ !# -  "  - œ !" #a b1 . It follows that . Hence there are no nontrivial solutions.- œ - œ !" # Suppose that . In this case, the general solution of the ODE is- .œ  # C B œ - -9=2 B  - =382 Ba b " #. . . The first boundary condition requires that . Imposing the second condition,- œ !" - -9=2  =382  - =382  -9=2 œ !" #a b a b.1 . .1 .1 . .1 . The two boundary conditions result in - >+82  œ !#a b.1 . . Since the solution of the equation is , we have .only >+82  œ ! œ ! - œ !.1 . . # Hence there are no nontrivial solutions. Let , with . Then the general solution of the ODE is- . .œ  !# C B œ - -9= B  - =38 Ba b " #. . . Imposing the boundary conditions, we obtain and- œ !" - -9=  =38  - =38  -9= œ !" #a b a b.1 . .1 .1 . .1 . For a solution of the ODE, we require that . Note thatnontrivial =38  -9= œ !.1 . .1 -9= œ ! Ê =38 œ !.1 .1 , which is false. It follows that . From a plot of and ,>+8 œ  >+8 .1 . 1 1. 1. —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 721 we find that there is a sequence of solutions, , , ; For large. ." #¸ !Þ()(' ¸ "Þ'("' â values of ,8 1 . 1 8 ¸ #8  " # a b . Therefore the eigenfunctions are , with corresponding eigenvalues9 .8 8a bB œ =38 B - -" #¸ !Þ'#!% ¸ #Þ(*%$ â Þ, , Asymptotically, -8 # ¸ #8  " % a b . 8. With , the general solution of the ODE is . Imposing the two- œ ! C B œ - B  -a b " # boundary conditions, and . It follows that . Hence- œ ! #-  - œ ! - œ - œ !" " # " # there are no nontrivial solutions. Setting , the general solution of the ODE is- .œ  # C B œ - -9=2 B  - =382 Ba b " #. . . The first boundary condition requires that . Imposing the second condition,- œ !# - -9=2  =382  - =382  -9=2 œ !" #a b a b. . . . . . . The two boundary conditions result in - "  >+82 œ !"a b. . . Since , it follows that , and there are no nontrivial solutions.. .>+82 ! - œ !" Let , with . Then the general solution of the ODE is- . .œ  !# C B œ - -9= B  - =38 Ba b " #. . . Imposing the boundary conditions, we obtain and- œ !# —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 724 Collecting the various terms, = ?  #=  #= ?  =  #=  "  = ? œ !ww w w ww wa b c da b- . The second term on the left vanishes as long as .= œ =w a b a b, = B œ /. With , the transformed differential equation can be written asB ?  ? œ !ww - . Since the boundary conditions are , we also have . It nowhomogeneous ? ! œ ? " œ !a b a b follows that the eigenfunctions are , with corresponding eigenvalues? œ =38 B8 8È- - 18 # #œ 8 . Therefore the eigenfunctions for the original problem are , with9 18 Ba bB œ / =388 B corresponding eigenvalues "  œ "  8- 18 # #. a b- . The given equation is a second order differential equation. Theconstant coefficient characteristic equation is <  #<  "  œ !# a b- , with roots .< œ "„ "ß# È - If , then the general solution is . Imposing the two boundary- œ ! C œ - /  - B/" #B B conditions, we find that , and hence there are no nontrivial solutions. If- œ - œ !" # -  ! , then the general solution is C œ - /B: B  - /B: B" #Š ‹ Š ‹È È"   "  - - . It again follows that , and hence there are no nontrivial solutions.- œ - œ !" # Therefore , and the general solution is-  ! C œ - / -9= B  - / =38 B" # B BÈ È- - . Invoking the boundary conditions, we have and . For a- œ ! - /=38 œ !" # È- nontrivial solution, .È- 1œ 8 19. First write the differential equation as C  "  C  C œ !ww wa b- - , which is a second order differential equation. The characteristicconstant coefficient equation is —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 725 <  "  <  œ !# a b- - , with roots and . For , the general solution is< œ  " < œ  Á "" # - - C œ - /  - /" # B  B- . Imposing the boundary conditions, we require that and .-  - œ ! - /  - / œ !" # " #" - For a nontrivial solution, it follows that , and hence , which is contrary/ œ / œ "" - - to the assumption. If , then the general solution is- œ " C œ - /  - B/" # B B. The boundary conditions require that and . Hence there are- œ ! - /  - / œ !" " #" " no nontrivial solutions. 21. Suppose that . In that case the general solution is . The- œ ! C œ - B  -" # boundary conditions require that and . We find that , and-  #- œ ! -  - œ ! - œ - œ !" # " # " # hence there are no nontrivial solutions. a b+ œ  !. Let , with . Then the general solution of the ODE is- . .# C B œ - -9= B  - =38 Ba b " #. . . The boundary conditions require that #-  - œ ! - -9=  - =38 œ !" # " #. . . and . These equations have a nonzero solution only if #=38  -9= œ !. . . , which can also be written as #>+8  œ !. . . Based on the graph, the positive roots of the determinantal equation are —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 726 . . . 1 " #¸ %Þ#(%) ¸ (Þ&*'& â 8 ¸ #8  " # , , ; for large , . 8 a b Therefore the eigenvalues are - - - 1 " #¸ ")Þ#($) ¸ &(Þ(!(& â 8 ¸ #8  " % , , ; for large , . 8 # #a b a b, œ   !. Setting , the general solution of the ODE is- .# C B œ - -9=2 B  - =382 Ba b " #. . . Imposing the boundary conditions, we obtain the equations #-  - œ ! - -9=2  - =382 œ !" # " #. . . and . These equations have a nonzero solution only if #=382  -9=2 œ !. . . . The latter equation is satisfied only for and . Hence the only. .œ ! œ „"Þ*"&! negative eigenvalue is .-" œ $Þ''($ 24. Based on the physical problem, . Let . The characteristic- = - .œ 7 ÎIM  ! œ# % equation is , with roots , and . Hence the<  œ ! < œ „ 3 < œ  < œ% %. . . ."ß# $ % general solution is C B œ - -9=2 B  - =382 B  - -9= B  - =38 Ba b " # $ %. . . . . a b a b a b a b a b+ À C ! œ C ! œ ! à C P œ C P œ ! Þ. Simply supported on both ends ww ww Invoking the boundary conditions, we obtain the system of equations -  - œ ! -  - œ ! - -9=2 P  - =382 P  - -9= P  - =38 P œ ! - -9=2 P  - =382 P  - -9= P  - =38 P œ ! " $ " $ " # $ % " # $ % . . . . . . . . . . . .# # # # . The determinantal equation is . . .%=382 P =38 P œ ! . The nonzero roots are , . The first two equations result in. 18 œ 8 ÎP 8 œ "ß #ßâ - œ - œ !" $ . The last two equations, - =382 8  - =388 œ ! - =382 8  - =388 œ ! # % # % 1 1 1 1 , imply that . Therefore the eigenfunctions are , with corresponding- œ ! œ =38 B# 9 .8 8 eigenvalues .- 18 % % %œ 8 ÎP —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 729 \ X \ I X œ ww ww3 . Since both sides of the resulting equation are functions of different variables, each must be equal to a constant, say . Therefore we obtain two ordinary differential equations - \  \ œ ! X  X œ ! Iww ww- - 3 and . a b a b a b a b a b, ? ! ß > œ \ ! X > >  ! \ ! œ !. Given that for , it follows that . The second boundary condition can be expressed as IE\ P X > 7\ P X > œ ! >  !w wwa b a b a b a b , . From the result in Part ,a b+ IE\ P X >  7 \ P X > œ ! >  ! Iwa b a b a b a b- 3 , . Since the condition is to be satisfied for all , we arrive at the boundary condition>  ! \ P  \ P œ ! 7 E wa b a b- 3 . a b- œ !. If , the general solution of the spatial equation is- \ B œ - B  -a b " # . The boundary condition require that . Hence there are no nontrivial- œ - œ !" # solutions. If , then the general solution is- .œ   !# \ B œ - -9=2 B  - =382 Ba b " #. . . The first boundary condition implies that . The second boundary condition- œ !" requires that - -9=2 P  - =382 P œ ! 7 E # #. . . 3 . The solution is nontrivial only if . >+82 P œ  E 7 . 3 . Since . >+82 P !. , there are no nontrivial solutions. Let - .œ  !# . The general solution of the spatial equation is \ B œ - -9= B  - =38 Ba b " #. . . —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 730 The first boundary condition implies that . The second boundary condition- œ !" requires that - -9= P  - =38 P œ ! 7 E # #. . . 3 . For a nontrivial solution, it is necessary that -9= P  =38 P œ ! 7 E . . . 3 , or >+8 P œ E 7 . 3 . . For the case ,a b7Î EP œ !Þ&3 we find that and . Therefore the eigenfunctions are given. ." #P ¸ "Þ!('* P ¸ $Þ'%$' by . The corresponding eigenvalues are solutions of9 .8 8a bB œ =38 B -9= P  =38 P œ ! P # È È È- - -8 8 8 . The first two eigenvalues are approximated as and .- -" #¸ "Þ"&*(ÎP ¸ "$Þ#('ÎP# # —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 731 Section 11.2 2. Based on the boundary conditions, . The general solution of the ODE is-  ! C B œ - -9= B  - =38 Ba b È È" #- - . The boundary condition requires that . Imposing the second boundaryC ! œ ! - œ !wa b # condition, we find that . So for a nontrivial solution, ,- -9= œ ! œ #8  " Î#" È È a b- - 1 8 œ "ß #ßâ Þ Therefore the eigenfunctions are given by 9 1 8 8a b a bB œ 5 -9= #8  " B # . In this problem, , and the normalization condition is< B œ "a b 5 -9= .B œ " #8  " B #8 # ! " #( ” •a b1 . It follows that . Therefore the normalized eigenfunctions are5 œ #8# 9 1 8a b È a bB œ # -9= 8 œ "ß #ßâ Þ#8  " B # , 3. Based on the boundary conditions, . For , the eigenfunction is- - ! œ ! 9! !a bB œ 5 Þ Set . With , the general solution of the ODE is5 œ "  !! - C B œ - -9= B  - =38 Ba b È È" #- - . Invoking the boundary conditions, we require that and .- œ ! - =38 œ !# "È È- - Since - - 1 ! œ 8 8 œ "ß #ßâ, the eigenvalues are , , with corresponding eigenfunctions8 # # 9 18 8a bB œ 5 -9= 8 B . The normalization condition is 5 -9= 8 B .B œ "8 # # ! "( 1 . It follows that . Therefore the normalized eigenfunctions are5 œ #8# 9 9 1!a b a b ÈB œ " B œ # -9= 8 B 8 œ "ß #ßâ Þ, and , 8 4. From Prob. in Section , ) ""Þ" the eigenfunctions are , in9 -8 8 8a b ÈB œ 5 -9= B which -9=  =38 œ !È È È- - -8 8 8 . The normalization condition is —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 734 in which the eigenvalues satisfy . Based on Eq. , -9=  =38 œ ! $%È È È a b- - -8 8 8 the coefficients in the eigenfunction expansion are given by - œ 0 B B .B œ B .B $  -9 œ # # 7 7 ! " ! " 7 7 ( a b a b ( È ˆ ‰ 9 - ! # = # -9= B -9=  " É È È È - - - 7 7 7 , in which !7 œ "  ÞÉ =38# 7È- 12. The normalized eigenfunctions are given by 9 - - 8 8 8 a b ÈÉ ÈB œ # -9= B = #$  -9 , in which the eigenvalues satisfy . Based on Eq. , -9=  =38 œ ! $%È È È a b- - -8 8 8 the coefficients in the eigenfunction expansion are given by - œ 0 B B .B œ "  B .B $  -9 œ # "  7 7 ! " ! " 7 7 ( a b a b ( a b È ˆ ‰ 9 - ! # = # -9= B -9= É È È È - - - 7 7 7 , in which !7 œ "  ÞÉ =38# 7È- 13. We consider the normalized eigenfunctions 9 - - 8 8 8 a b ÈÉ ÈB œ # -9= B = #$  -9 , in which the eigenvalues satisfy . T-9=  =38 œ !È È È- - -8 8 8 he coefficients in the eigenfunction expansion are given by —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 735 - œ 0 B B .B œ .B $  -9 œ # 8 8 ! " ! "Î# 8 ( a b a b ( È 9 ! # = # -9= B =38 Î# É È È ˆ ‰ÈÈ - - - - 8 8 8 8 , in which !8 œ "  ÞÉ =38# 8È- 15. The differential equation can be written as  ‘ˆ ‰"  B C  C œ !# w w , with and . The boundary conditions are homogeneous and: B œ  "  B ; B œ "a b a b# separated self-adjoint. Hence the BVP is . 16. Since the boundary conditions are separated, the inner product is computed:not Given and , sufficiently smooth and satisfying the boundary conditions,? @ a b c dc d ( ¹ ( c d c d a b¹ c d P ? ß @ œ ? @  ?@ .B œ ? @  ? @  ?@ .B œ ? @  ?@  ?ßP @ Þ ! " ww w w w ! " ! " w w ! " Based on the given boundary conditions, ? " @ "  ? ! @ ! œ ? ! @ "  #? " @ !  ? " @ "  ? ! @ ! œ  ? " @ !  #? ! @ " Þ w w w w a b a b a b a b a b a b a b a ba b a b a b a b a b a b a b a b Since c d a b a b a b a b¹? @  ?@ œ ? " @ !  ? ! @ "w w ! " , the BVP is self-adjoint.not 18. The differential equation can be written as  C œ Cc dw w - , with , , and . The boundary conditions are homogeneous: B œ " ; B œ ! < B œ "a b a b a b and separated self-adjoint. Hence the BVP is . —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 736 19. If , then+ œ !# ? " @ "  ? " @ " œ  ? " @ "  ? " @ " œ ! , , , , w w w w w wa b a b a b a b a b a b a b a b# # " " , and since ,? ! œ @ ! œ !a b a b ? ! @ !  ? ! @ ! œ !w wa b a b a b a b . If , then implies that, œ ! ? " œ @ " œ !# a b a b ? " @ "  ? " @ " œ !w wa b a b a b a b . Furthermore, ? ! @ !  ? ! @ ! œ  ? ! @ !  ? ! @ ! œ ! + + + + w w w w w wa b a b a b a b a b a b a b a b# # " " . Clearly, the results are also true if .+ œ , œ !# # 20. Suppose that and are linearly independent eigenfunctions associated9 9" #a b a bB B with an eigenvalue . The Wronskian is given by- [ B œ B B  B Ba ba b a b a b a b a b9 9 9 9 9 9" # " ## ", .w w Each of the eigenfunctions satisfies the boundary condition . If+ C !  + C ! œ !" #a b a bw either or , then clearly , . On the other hand, if is+ œ ! + œ ! [ ! œ ! +" # " # #a ba b9 9 not equal to zero, then [ ! œ ! !  ! ! œ  ! !  ! ! + + + + œ ! a ba b a b a b a b a b a b a b a b a b 9 9 9 9 9 9 9 9 9 9 " # " ## " " " # # " # # " , . w w By Theorem , , for all . Based on Theorem ,$Þ$Þ# [ B œ ! ! Ÿ B Ÿ " $Þ$Þ$a ba b9 9" # 9 9 -" #a b a bB B and must be linearly . Hence must be a simple eigenvalue.dependent 22. We consider the operator P C œ  : B C  ; B Cc d c d a ba b w w on the interval , together with the boundary conditions!  B  " + C !  + C ! œ ! , C "  , C " œ !" # " #a b a b a b a bw w , . Let and . If and both satisfy the boundary conditions, then? œ  3 @ œ  3 ? @9 < 0 ( the real and imaginary parts also satisfy the same boundary conditions. Using the inner product a b a b a b(? ß @ œ ? B @ B .B ! " , —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 739 9 -8 8a b ÈB œ "  -9= B , and the eigenvalues satisfy the equation #  #-9= P œ P=38 P ÞÈ È È- - -8 8 8 The smallest eigenvalue is - 1" œ # ÎP Þa b# # 26. As shown is Prob. , the general solution is#& C B œ -  - B  - -9= B  - =38 Ba b " # $ %. . . Imposing the boundary conditions, we obtain the system of equations - œ ! -  - œ ! -  - œ ! - -9= P  - =38 P œ ! # " $ # % $ % . . . . For a nontrivial solution, it is necessary that -9= P œ !. . We find that , and hence the eigenfunctions are given by- œ - œ !# % 9 -8 8a b ÈB œ "  -9= B . The corresponding eigenvalues are , . The smallest- 18 # # #œ #8  " Î%P 8 œ "ß #ßâa b eigenvalue is - 1" œ Î%P Þ# # —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 740 Section 11.3 4. The eigensystem of the associated homogeneous problem is given in Prob. of"" Section . The normalized eigenfunctions are""Þ# 9 - - 8 8 # 8 a b È ÈÉ ÈB œ # -9= B "  =38 , in which the eigenvalues satisfy . Rewrite the given-9=  =38 œ !È È È- - -8 8 8 differential equation as . Since , the formal solution of C œ #C  B œ # Áww 8. - the nonhomogeneous problem is C B œ B -  # a b a b" 8œ" _ 8 8 8 - 9 , in which - œ 0 B B .B œ B .B œ # # 8 8 ! " ! " 8 ( a b a b ( È ˆ ‰ 9 - È É È # "  =38# 8- -9= B -9=  " "  =38 È È É È - - - 8 8 # 8 . Therefore we obtain the formal expansion C B œ # # #  # a b "È ˆ ‰a bˆ ‰8œ" _ 8 8 -9=  " -9= B "  =38 È È È- --8 8# 8- - . 5. The solution follows that in Prob. , except that the coefficients are given by" - œ 0 B B .B œ # #B .B  # #  #B .B œ % # =38 8 Î# 8 8 8 ! " ! "Î# "Î# " # # ( a b a b È È( ( a b È a b 9 1 1 =388 B =388 B1 1 . Therefore the formal solution is C B œ ) =38 8 Î# 8 8  # a b " a ba b 8œ" _ # # # # 1 1 1 =388 B1 . —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 741 6. The differential equation can be written as  C œ C  0 B ; B œ !ww . a b a b. Note that and . As shown in Prob. in Section , t< B œ " " ""Þ#a b he normalized eigenfunctions are 98a b È a bB œ # =38 #8  " B # , with associated eigenvalues . Based on Theorem , the- 18 # #œ #8  " Î% ""Þ$Þ"a b formal solution is given by C B œ -  a b "a bÈ a b# =38 #8  " B# 8œ" _ 8 -8 . , as long as The coefficients in the series expansion are computed as. -Á Þ8 - œ 0 B8 ! "È a b# =38 .B#8  " B # ( a b . 7. he normalized eigenfunctions areAs shown in Prob. in Section , t" ""Þ# 98a b È a bB œ # -9= #8  " B # , with associated eigenvalues . Based on Theorem , the- 18 # #œ #8  " Î% ""Þ$Þ"a b formal solution is given by C B œ -  a b "a bÈ a b# -9= #8  " B# 8œ" _ 8 -8 . , as long as The coefficients in the series expansion are computed as. -Á Þ8 - œ 0 B -98 ! "È a b# = .B#8  " B # ( a b . 9. The normalized eigenfunctions are 9 - - 8 8 # 8 a b È ÈÉ ÈB œ # -9= B "  =38 . The eigenvalues satisfy . Based on Theorem , the-9=  =38 œ ! ""Þ$Þ"È È È- - -8 8 8 formal solution is given by C B œ -  a b "a bÈ È É È# -9= B "  =388œ" _ 8 - - - 8 8 8 #. , as long as The coefficients in the series expansion are computed as. -Á Þ8 —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 744 Substitution into the given PDE results in " "a b a b a b a b " a b a b 8œ" 8œ" _ _ 8 8 w ww > 8 8 8œ" _ 8 8 8 > , > B œ , > B  / œ  , > B  / 9 9 - 9 , that is, "c d a ba b a b 8œ" _ 8 w > 8 8 8, >  , > B œ / Þ- 9 We now note that " œ B # =38 "  =38 " È ÈÈ ÈÉ a b8œ" _ 8 8 8 # 8 - - - 9 . Therefore / œ / B> > 8œ" _ 8 8" a b" 9 , in which . Combining these results," - - -8 8 8 8#œ # =38 Î "  =38È È È È” •É " ‘a b a b a b 8œ" _ 8 w > 8 8 8 8, >  , >  / B œ ! Þ- " 9 Since the resulting equation is valid for , it follows that!  B  " , >  , > œ / 8 œ "ß #ßâ8 w > 8 8 8a b a b- " , . Prior to solving the sequence of ODEs, we establish the initial conditions. These are obtained from the expansion ? B ß ! œ "  B œ Ba b a b" 8œ" _ 8 8! 9 , in which That is, .! - - - !8 8 8 8 8 8#œ # "  -9= Î "  =38 Þ , ! œÈ ˆ ‰È È” •É a b Therefore the solutions of the first order ODEs are , > œ  / 8 œ "ß #ßâ /  /  " 8 8 8 >  > 8  >a b ˆ ‰a b" - ! - - 8 8 , . Hence the solution of the boundary value problem is —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 745 ? B ß > œ  / B Þ /  /  " a b a b"– —ˆ ‰a b 8œ" _ 8 >  > 8 8 8  >" - ! 9 - - 8 8 21. Based on the boundary conditions, the normalized eigenfunctions are given by 9 18a b ÈB œ # =388 B , with associated eigenvalues We now assume a solution of the form- 18 # #œ 8 Þ ? B ß > œ , > Ba b a b a b" 8œ" _ 8 89 . Substitution into the given PDE results in " "a b a b a b a b k k " a b a b k k 8œ" 8œ" _ _ 8 8 w ww 8 8 8œ" _ 8 8 8 , > B œ , > B  "  "  #B œ  , > B  "  "  #B 9 9 - 9 , that is, "c d a b k ka b a b 8œ" _ 8 w 8 8 8, >  , > B œ "  "  #B Þ- 9 It was shown in Prob. that& "  "  #B œ Bk k a b" 8œ" _ 8% # =38 8 Î# 8 È a b1 1# # 9 . Substituting on the right hand side and collecting terms, we obtain "– —a b a b a b 8œ" _ 8 w 8 8 8, >  , >  B œ !Þ- 9% # =38 8 Î# 8 È a b1 1# # Since the resulting equation is valid for , it follows that!  B  " , >  , > œ 8 œ "ß #ßâ8 w 8a b a b8 % # =38 8 Î# 8 # # # # 1 1 1 È a b , . Based on the given initial condition, we also have , for . The, ! œ ! 8 œ "ß #ßâ8a b solutions of the first order ODEs are , > œ / 8 œ "ß #ßâ8 8 >a b % " # =38 8 Î# 8 È a bŠ ‹1 1% % # #1 , . Hence the solution of the boundary value problem is —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 746 ? B ß > œ / =388 B Þ )a b " 1 1 % 8œ" _ 8 >=38 8 Î# 8 "  a bŠ ‹1 % # #1 23 . Let be a solution of the boundary value problem and be a solutiona b a b a b+ ? B ß > @ B of the related BVP. Substituting for , we have? B ß > œ A B ß >  @ Ba b a b a b < B ? œ < B Aa b a b> > and c d a b a b c d a b c d a b a ba b a b a bc d a b a b a ba bc d a ba b : B ?  ; B ?  J B œ : B A  ; B A  : B @  ; B @  J B œ : B A  ; B A  J B  J B œ : B A  ; B A B BB B w w B B B B . Hence is a solution of the PDEA B ß >a b homogeneous < B A œ : B A  ; B Aa b c d a ba b> B B . The required areboundary conditions A ! ß > œ ? ! ß >  @ ! œ ! A " ß > œ ? " ß >  @ " œ ! a b a b a ba b a b a b ,. The associated is .initial condition A B ß ! œ ? B ß !  @ B œ 0 B  @ Ba b a b a b a b a b a b a b, @ B. Let be a solution of the ODE c d a b a ba b: B @  ; B @ œ  J Bw w , and satisfying the boundary conditions , .@ !  2 @ ! œ X @ "  2 @ " œ Xw w" #a b a b a b a b" # If , then it is easy to show the satisfies the PDE and initialA B ß > œ ? B ß >  @ B Aa b a b a b condition given in Part . Furthermore,a b+ A ! ß >  2 A ! ß > œ ? ! ß >  @ !  2 ? ! ß >  2 @ ! œ ? ! ß >  2 ? ! ß >  @ !  2 @ ! œ ! B B w B w a b a b a b a b a b a ba b a b a b a b" " "" " . Similarly, the other boundary condition is also homogeneous. —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 749 # 9 9 8 8 ! " ! " 8 a b a b a b( a ba b ( a b a b > œ < B B .B J B ß > < B œ J B ß > B .B 8 œ "ß #ßâ, . Combining these results, we have "c da b a b a b 8œ" _ 8 ww 8 8 8 8, >  , >  > œ !- # 9 . It follows that , >  , > œ > 8 œ "ß #ßâ8 ww 8 8 8a b a b a b- # , . In order to solve this sequence of ODEs, we require initial conditions and , ! , ! Þ8 8wa b a b Note that ? B ß ! œ , ! B ? B ß ! œ , ! Ba b a b a b a b a b a b" " 8œ" 8œ" _ _ 8 8 > 88 w9 9 and . Based on the given initial conditions, 0 B œ , ! B 1 B œ , ! Ba b a b a b a b a b a b" " 8œ" 8œ" _ _ 8 8 88 w9 9 and . Hence and , the expansion coefficients for and in, ! œ , ! œ 0 B 1 B8 8 88wa b a b a b a b! " terms of the eigenfunctions, .98a bB 27 . Since the eigenvectors are , they form a basis. Given any vector ,a b+ orthogonal b b œ , Þ" 3œ " 8 30 a b3 Taking the inner product, with , of both sides of the equation, we have0a b4 ˆ ‰ ˆ ‰b ß œ , ß Þ0 0 0a b a b a b4 4 44 a b, . Consider solutions of the form x œ + Þ" 3œ " 8 30 a b3 Substituting into Eq. , and using the above form of ,a b3 b " " " 3œ " 3œ" 3œ" 8 8 8 3 3 3+  + œ , ÞA0 0 0a b a b a b3 3 3. It follows that —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 750 "c d 3œ " 8 3 3 3 3+  +  , œ Þ- . 0 a b3 0 Since the eigenvectors are linearly independent, +  +  , œ ! 3 œ "ß #ßâß 83 3 3 3- . , for . That is, + œ , Î  3 œ "ß #ßâß 83 3 3a b- . , . Assuming that the eigenvectors are , the solution is given bynormalized x b œ ß  "a b 3œ " 8 3 0 0 a b a b3 3 - . , as long as is equal to one of the eigenvalues.. not 29. First write the ODE as . A fundamental set of solutions of theC  C œ  0 Bww a b homogeneous equation is given by C œ -9= B C œ =38 B" # and . The Wronskian is equal to . Applying the method of [ -9= B ß =38 B œ "c d variation of parameters, a particular solution is ] B œ C B ? B  C B ? Ba b a b a b a b a b" " # # , in which ? B œ =38 = 0 = .= ? B œ  -9= = 0 = .=" #a b a b a b a b a b a b( ( ! ! B B and . Therefore the general solution is C œ B œ - -9= B  - =38 B  -9= B =38 = 0 = .=  =38 B -9= = 0 = .=9a b a b a b a b a b( (" # ! ! B B . Imposing the boundary conditions, we must have and- œ !" - =38 "  -9= " =38 = 0 = .=  =38 " -9= = 0 = .= œ !# ( (a b a b a b a b ! ! " " . It follows that - œ =38 "  = 0 = .= " =38 " # ( a b a b ! " , and —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 751 9a b a b a b a b a b( (B œ =38 "  = 0 = .=  =38 B  = 0 = .==38 B =38 " ! ! " B . Using standard identities, =38 B † =38 "  =  =38 " † =38 B  = œ =38 = † =38 "  Ba b a b a b . Therefore =38 B † =38 "  = =38 = † =38 "  B =38 " =38 "  =38 B  = œ a b a ba b . Splitting up the integral, we obtainfirst 9a b a b a b( (a b a b ( a b a b B œ 0 = .=  0 = .= =38 = † =38 "  B =38 B † =38 "  = =38 " =38 " œ K B ß = 0 = .= ! B B " ! " , in which K B ß = œ ß ! Ÿ = Ÿ B ß B Ÿ = Ÿ " Þ a b  =38 =†=38 "B =38 " =38 B†=38 "= =38 " a b a b 31. The general solution of the homogeneous problem is C œ -  - B" # . By inspection, it is easy to see that satisfies the BC and that theC B œ " C ! œ !"a b a bw function satisfies the BC . The Wronskian of these solutions isC B œ "  B C " œ !#a b a b [ C ß C œ  " $! : B œ "c d a b" # . Based on Prob. , with , the Green's function is given by K B ß = œ "  B ß ! Ÿ = Ÿ B "  = ß B Ÿ = Ÿ " Þ a b œ a ba b Therefore the solution of the given BVP is 9a b a b a b a b a b( (B œ "  B 0 = .=  "  = 0 = .= ! B B " . 32. The general solution of the homogeneous problem is C œ -  - B" # . We find that satisfies the BC . Imposing the boundary conditionC B œ B C ! œ !"a b a b —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 754 9 - . 93 35 5 5 5 œ" _a b a b a b"B œ ,  B . Since the eigenfunctions are linearly independent, , and thus,  œ35 5 35a b- . $ C B œ B œ B   " 3a b a b a b" 5 œ" _ 35 5 3 5 3 $ - . - . 9 9 . We conclude that + B ß œ B "  3 3 3 a b a b. 9 - . , which verifies that K B ß = ß œ B =  a b " a b a b. 9 9 - . 3œ " _ 3 3 3 . 36. First note that for . On the interval , the solution . CÎ.= œ ! = Á B !  =  B# # of the ODE is . Given that , we have . On theC = œ -  - = C ! œ ! C = œ - =" " # " #a b a b a b interval , the solution is . Imposing the condition ,B  =  " C = œ .  . = C " œ !# " #a b a b we have . Assuming continuity of the solution, at ,C = œ . "  = = œ B# "a b a b - B œ . "  B# "a b , which gives . Next, integrate both sides of the given ODE over an- œ . "  B ÎB# "a b infinitesimal interval containing = œ B À  .= œ =  B .= œ " . C .= ( ( a b B B B B# #     $ . It follows that C B  C B œ "w  w a b a b , and hence . Solving for the two coefficients, we obtain and-   . œ " - œ "  B# " #a b . œ B" . Therefore the solution of the BVP is given by C = œ = "  B ß ! Ÿ = Ÿ B B "  = ß B Ÿ = Ÿ " a b œ a ba b , which is identical to the Green's function in Prob. .#) —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 755 Section 11.4 1. Let be the eigenfunctions of the singular problem9 -8 8a b ˆ ‰ÈB œ N B!  B C œ BC !  B  " C ß C Bp! C " œ ! a b a b w w w - , , bounded as , . Let be a solution of the given BVP, and set9a bB 9 9a b a b a b"B œ , B ‡ 8œ! _ 8 8 . Then  B œ B  0 B œ B  B 0 B B a b a ba b9 . 9 . 9 w w . Substituting , we obtaina b‡ " " "a b a b a b 8œ! 8œ! 8œ! _ _ _ 8 8 8 8 8 8 8, B B œ B , B  B - B- 9 . 9 9 , in which the are the expansion coefficients of for . That is,- 0 B ÎB B  !8 a b - œ B B .B " 0 B m B m B œ 0 B B .B " m B m 8 8 8 # ! " 8 # ! " 8 9 9 9 9 a b ( a b a b a b ( a b a b . It follows that if ,B Á ! "c d a ba b 8œ! _ 8 8 8 8-  ,  B œ !- . 9 . As long as , linear independence of the eigenfunctions implies that. -Á 8 , œ 8 œ "ß #ßâ -  8 8 8- . , . Therefore a formal solution is given by 9 - - . a b " Š ‹ÈB œ N B-  8œ! _ 8 8 8! , in which are the positive roots of .È a b-8 N B œ !! —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 756 3 . Setting , it follows thata b È+ > œ B- .C .C . C . C .B .> .B .> œ œÈ- - and .# # # # The given ODE can be expressed as   œ > C . > .C 5 .> .> > È È È È È - - - - -# , or  >  œ > C Þ . .C 5 .> .> > Œ  # An equivalent form is given by >  >  >  5 C œ ! .C .C .> .> # # #ˆ ‰ , which is known as a Bessel equation of order . A solution is .5 N >bounded 5a b a b Š ‹È, N B B œ !. satisfies the boundary condition at . Imposing the other5 - boundary condition, it is necessary that . Therefore the eigenvalues are given byN œ !5Š ‹È- - -8 8 5, , where are the positive zeroes of . The eigenfunctions of8 œ "ß #â N BÈ a b the BVP are 9 -8 5 8a b ˆ ‰ÈB œ N B Þ a b- . The BVP is a singular Sturm-Liouville problem with P C œc d  BC  C < B œ "5 B a b a bw w # and . We note that - - 8 ! ! " " ! " 7 ! " ( ( ( ( B B 9 9 9 9 9 9 9 9 8 7 8 7 8 7 8 7 a b a b c d a b a b c d a b a b B B .B œ P B .B œ B P .B œ B B .B Þ Therefore a b(- -8 7 ! "  B9 98 7a b a bB B .B œ ! Þ —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 759 8 "  B "  B # " " " " # " " # " " # ( (È ( ( È X B X B .B œ P X X B .B œ X B P X .B œ 7 .B Þ X B X B 8 7 8 7 8 7 8 7 a b a b c d a b a b c d a b a b Therefore ˆ ‰( È8 7 "  B# # " " # X B X B .B œ ! Þ 8 7a b a b So for ,8 Á 7 ( È" " # X B X B .B œ ! Þ 8 7a b a b "  B —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 760 Section 11.5 3. The equations relating to this problem are given by Eqs. to in the text. Baseda b a b# "( on the boundary conditions, the eigenfunctions are and the associated9 -8 8a b a bB œ N <! eigenvalues are the positive zeroes of . The general solution has the- - -" # !ß ßâ N a b form ? < ß > œ - N < -9= +>  5 N < =38 +>a b c d" a b a b 8œ" _ 8 8 8 8 8 8! !- - - - . The initial conditions require that ? < ß ! œ - N < œ 0 <a b a b a b" 8œ" _ 8 8! - and ? < ß ! œ + 5 N < œ 1 <> 8 8 8 8œ" _a b a b a b" - -! . The coefficients and are obtained from the respective eigenfunction expansions.- 58 8 That is, - œ <0 < N < .< " mN < m 8 8 8 # ! " ! !a b ( a b a b- - and 5 œ <1 < N < .< " + mN < m 8 8 8 8 # ! " - - - ! !a b ( a b a b , in which mN < m œ < N < .<! !a b c d( a b- -8 8# ! " # for .8 œ "ß #ß â 8. A more general equation was considered in Prob. of Section . #$ "!Þ& Assuming a solution of the form , substitution into the PDE results in? < ß > œ V < X >a b a b a b !# ww w w” •V X  V X œ VX" < . Dividing both sides of the equation by the factor , we obtainVX —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 761 V " V X V < V X  œ ww w w #! . Since both sides of the resulting differential equation depend on variables, eachdifferent side must be equal to a constant, say . That is, -# V " V X V < V X  œ œ  ww w w # # ! - . It follows that , andX  X œ !w # #! - V " V V < V  œ  ww w #- , which can be written as . Introducing the variable ,< V  <V  < V œ ! œ <# ww w # #- 0 - the last equation can be expressed as , which is the Bessel0 0 0# ww w #V  V  V œ ! equation of order zero. The temporal equation has solutions which are multiples of . TheX > œ /B:a b a b >! -# # general solution of the Bessel equation is V < œ , , ]a b " #N <  <! !a b a b- -8 8 . Since the steady state temperature will be , all solutions must be bounded, and hencezero we set . Furthermore, the boundary condition requires that , œ ! ? " ß > œ ! V " œ !# a b a b and hence N œ ! B œ N <! !a b a b a b- 9 -. It follows that the eigenfunctions are , with the8 8 associated eigenvalues , which are the positive zeroes of .- - -" # !ß ßâ N a b Therefore the fundamental solutions of the PDE are , and the? < ß > œ /B:8a b N <!a b-8 a b >! -# #8 general solution has the form ? < ß > œ - N <a b a b" 8œ" _ 8 8! - /B:  >ˆ ‰! -# #8 . The initial condition requires that ? < ß ! œ - N < œ 0 <a b a b a b" 8œ" _ 8 8! - . The coefficients in the general solution are obtained from the eigenfunction expansion of 0 <a b. That is, - œ <0 < N < .< " mN < m 8 8 8 # ! " ! !a b ( a b a b- - , in which mN < m œ < N < .< 8 œ "ß #ß â! !a b c d a b( a b- -8 8# ! " # . —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 764 T œ ! œ B ß B ßâß B œ "R " # R"e f be a of into equal subintervals. We can always choose a point,partition rationalc d! ß " 03 , in each of the subintervals so that the Riemann sum V ß ßâß œ 0 œ " " R a b a b"0 0 0 0" # R 8œ" R 8 . Likewise, can always choose an point, , in each of the subintervals so thatirrational (3 the Riemann sum V ß ßâß œ 0 œ  " " R a b a b"( ( ( (" # R 8œ" R 8 . It follows that is Riemann integrable.0 Ba b not 8. With and , the normalization conditions are satisfied. UsingT B œ " T B œ B! "a b a b the usual inner product on ,c d " ß " ( a b a b " " T B T B .B œ !! " and hence the polynomials are also orthogonal Let . TheÞ T B œ + B  + B  +# # " !a b # normalization condition requires that . For orthogonality, we need+  +  + œ "# " ! ( (ˆ ‰ ˆ ‰ " " " " # #+ B  + B  + .B œ ! B + B  + B  + .B œ !# " ! # " ! and . It follows that , and . Hence .+ œ $Î# + œ ! + œ  "Î# T B œ $B  " Î## " ! #a b a b# Now let . The coefficients must be chosen so thatT B œ + B  + B  + B  +$ $ # " !a b $ # +  +  +  + œ "$ # " ! and the orthogonality conditions ( a b a b a b " " 3 4T B T B .B œ ! 3 Á 4 are satisfied. Solution of the resulting algebraic equations leads to , ,+ œ &Î# + œ !$ # + œ  $Î# + œ ! T B œ &B  $B Î#1 and . Therefore .! $a b a b$ 11. The implied sequence of coefficients is , . Since the limit of these+ œ " 8 "8 coefficients is zero, the series cannot be an eigenfunction expansion.not 13. Consider the eigenfunction expansion 0 B œ + B Þa b a b" 3œ " _ 3 39 Formally, —————————————————————————— ——CHAPTER 11. ________________________________________________________________________ page 765 0 B œ + B  # + + B B Þ# # # 3œ" _ 3 3 3Á4 3 4 3 4a b a b a b a b" "9 9 9 Integrating term-by-term, ( ( (a b a b a b a b a b a b a b" " " ( a b ! ! ! " " " # # # 3œ" _ 3 3 3Á4 3 4 3 4 3œ " _ 3 3 # # ! " < B 0 B .B œ + < B B .B  # + + < B B B .B œ + B .B 9 9 9 9 , since the eigenfunctions are orthogonal. Assuming that they are also normalized, ( a b a b " ! " # # 3œ" _ 3< B 0 B .B œ + .