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# 13.1E: Boundary Value Problems (Exercises)

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University
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## Q13.1.1

1. Verify that $$B_{1}$$ and $$B_{2}$$ are linear operators; that is, if $$c_{1}$$ and $$c_{2}$$ are constants then $B_{i}(c_{1}y_{1}+c_{2}y_{2})=c_{1}B_{i}(y_{1})+c_{2}B_{i}(y_{2}),\quad i=1,2.\nonumber$

## Q13.1.2

In Exercises 13.1.2-13.1.7 solve the boundary value problem.

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

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 $y=\int_{a}^{b} G(x,t)F(t)\,dt +\frac{k_{2}}{B_{2}}(y_{1})y_{1} +\frac{k_{1}}{B_{1}(y_{2})}y_{2}.\nonumber$