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13.1E: Boundary Value Problems (Exercises)

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Q13.1.1

1. Verify that B1 and B2 are linear operators; that is, if c1 and c2 are constants then Bi(c1y1+c2y2)=c1Bi(y1)+c2Bi(y2),i=1,2.

Q13.1.2

In Exercises 13.1.2-13.1.7 solve the boundary value problem.

2. y, y(0)=-2,y(1)=1

3. y''=2-3x, y(0)=0,y(1)-y'(1)=0

4. y''-y=x, y(0)+y'(0)=3,y(1)-y'(1)=2

5. y''+4y=1, y(0)=3, y(\pi/2)+y'(\pi/2)=-7

6. y''-2y'+y=2e^{x}, y(0)-2y'(0)=3, y(1)+y'(1)=6e

7. y''-7y'+12y=4e^{2x}, y(0)+y'(0)=8, y(1)=-7e^{2} (see Example 13.1.5)

Q13.1.3

8. State a condition on F such that the boundary value problem y''=F(x), \quad y(0)=0, \quad y(1)-y'(1)=0\nonumber has a solution, and find all solutions.

9.

  1. State a condition on a and b such that the boundary value problem y''+y=F(x),\quad y(a)=0,\quad y(b)=0 \tag{A} has a unique solution for every continuous F, and find the solution by the method used to prove Theorem 13.1.3
  2. In the case where a and b don’t satisfy the condition you gave for (a), state necessary and sufficient on F such that (A) has a solution, and find all solutions by the method used to prove Theorem 13.1.4.

10. Follow the instructions in Exercise 13.1.9 for the boundary value problem y''+y=F(x),\quad y(a)=0,\quad y'(b)=0.\nonumber

11. Follow the instructions in Exercise 13.1.9 for the boundary value problem y''+y=F(x),\quad y'(a)=0,\quad y'(b)=0.\nonumber

Q13.1.4

In Exercises 13.1.12-13.1.15 find a formula for the solution of the boundary problem by the method used to prove Theorem 13.1.3. Assume that a<b.

12. y''-y=F(x),y(a)=0,y(b)=0

13. y''-y=F(x),y(a)=0,y'(b)=0

14. y''-y=F(x),y'(a)=0,y'(b)=0

15. y''-y=F(x),y(a)-y'(a)=0,y(b)+y'(b)=0

Q13.1.5

In Exercises 13.1.16-13.1.19 find all values of \omega such that boundary problem has a unique solution, and find the solution by the method used to prove Theorem 13.1.3. For other values of \omega , find conditions on F such that the problem has a solution, and find all solutions by the method used to prove Theorem 13.1.4.

16. y''+ \omega^{2}y=F(x),y(0)=0,y(\pi)=0

17. y''+ \omega^{2}y=F(x),y(0)=0,y'(\pi)=0

18. y''+ \omega^{2}y=F(x),y'(0)=0,y(\pi)=0

19. y''+ \omega^{2}y=F(x),y'(0)=0,y'(\pi)=0

Q13.1.6

20. Let \{z_{1},z_{2}\} be a fundamental set of solutions of Ly=0. Given that the homogeneous boundary value problem Ly=0, \quad B_{1}(y)=0,\quad B_{2}(y)=0\nonumber has a nontrivial solution, express it explicity in terms of z_{1} and z_{2}.

21. If the boundary value problem has a solution for every continuous F, then find the Green’s function for the problem and use it to write an explicit formula for the solution. Otherwise, if the boundary value problem does not have a solution for every continuous F, find a necessary and sufficient condition on F for the problem to have a solution, and find all solutions. Assume that a<b.

  1. y''=F(x), y(a)=0,y(b)=0
  2. y''=F(x), y(a)=0,y'(b)=0
  3. y''=F(x), y'(a)=0,y(b)=0
  4. y''=F(x), y'(a)=0,y'(b)=0

22. Find the Green’s function for the boundary value problem y''=F(x), \quad y(0)-2y'(0)=0, \quad y(1)+2y'(1)=0. \tag{A} Then use the Green’s function to solve (A) with

  1. F(x)=1,
  2. F(x)=x, and
  3. F(x)=x^{2}.

23. Find the Green’s function for the boundary value problem x^{2}y''+xy'+(x^{2}-1/4)y=F(x), \quad y(\pi/2)=0,\quad y(\pi)=0, \tag{A} given that y_{1}(x)=\frac{\cos x}{\sqrt{x}}\quad \text{and}\quad y_{2}(x)=\frac{\sin x}{\sqrt{x}}\nonumber are solutions of the complementary equation. Then use the Green’s function to solve (A) with

  1. F(x)=x^{3/2} and
  2. F(x)=x^{5/2}.

24. Find the Green’s function for the boundary value problem x^{2}y''-2xy'+2y=F(x), \quad y(1)=0,\quad y(2)=0, \tag{A} given that \{x,x^{2}\} is a fundamental set of solutions of the complementary equation. Then use the Green’s function to solve (A) with

  1. F(x)=2x^{3} and
  2. F(x)=6x^{4}.

25. Find the Green’s function for the boundary value problem x^{2}y''+xy'-y=F(x), \quad y(1)-2y'(1)=0,\quad y'(2)=0, \tag{A} given that \{x,1/x\} is a fundamental set of solutions of the complementary equation. Then use the Green’s function to solve (A) with

  1. F(x)=1,
  2. F(x)=x^{2}, and
  3. F(x)=x^{3}.

Q13.1.7

In Exercises 13.1.26-13.1.30 find necessary and sufficient conditions on \alpha , β, ρ, and δ for the boundary value problem to have a unique solution for every continuous F, and find the Green’s function.

26. y''=F(x), \alpha y(0)+\beta y'(0)=0, \rho y(1)+\delta y'(1)=0

27. y''+y=F(x), \alpha y(0)+\beta y'(0)=0, \rho y(\pi)+\delta y'(\pi)=0

28. y''+y=F(x), \alpha y(0)+\beta y'(0)=0, \rho y(\pi/2)+\delta y'(\pi/2)=0

29. y''-2y'+2y=F(x), \alpha y(0)+\beta y'(0)=0, \rho y(\pi)+\delta y'(\pi)=0

30. y''-2y'+2y=F(x), \alpha y(0)+\beta y'(0)=0, \rho y(\pi/2)+\delta y'(\pi/2)=0

Q13.1.8

31. Find necessary and sufficient conditions on \alpha, \beta, \rho, and \delta for the boundary value problem y''-y=F(x), \quad \alpha y(a)+\beta y'(a)=0, \quad \rho y(b)+\delta y'(b)=0 \tag{A} to have a unique solution for every continuous F, and find the Green’s function for (A). Assume that a<b.

32. Show that the assumptions of Theorem 13.1.3 imply that the unique solution of Ly=F, \quad B_{1}(y)=k_{1},\quad B_{2}(y)=f_{2}\nonumber is y=\int_{a}^{b} G(x,t)F(t)\,dt +\frac{k_{2}}{B_{2}}(y_{1})y_{1} +\frac{k_{1}}{B_{1}(y_{2})}y_{2}.\nonumber


This page titled 13.1E: Boundary Value Problems (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.

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