Elementary Differential Equations 8th edition by Boyce ch11

Elementary Differential Equations, with ODE Architect CD

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—————————————————————————— —— 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.

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—————————————————————————— —— 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 ! " # 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 B a b " # . . . Imposing the boundary conditions, we obtain and - œ ! #
—————————————————————————— —— CHAPTER 11. ________________________________________________________________________ page 722 - -9= =38  - =38 -9= œ ! " # a b a b . . . . . . . For a solution of the ODE, we require that . First note nontrivial -9= =38 œ ! . . . that -9= œ ! Ê œ ! =38 œ ! . . . or . Therefore we find that . From a plot of , there is a sequence of "  >+8 œ ! >+8 . . . .

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