13.1E: Boundary Value Problems (Exercises)

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