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**Unformatted text preview: **—————————————————————————— —— 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 is nonzero 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 b 1 . 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 B a 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 B a 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 that nontrivial =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 eigenvalues 9 . 8 8 a b B 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 B a b " # . . . The first boundary condition requires that . Imposing the second condition,- œ ! #--9=2 =382 - =382 -9=2 œ !...

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