13.1E: Boundary Value Problems (Exercises)

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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.

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
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 $a<b$.

$y''=F(x)$, $y(a)=0$,$y(b)=0$
$y''=F(x)$, $y(a)=0$,$y'(b)=0$
$y''=F(x)$, $y'(a)=0$,$y(b)=0$
$y''=F(x)$, $y'(a)=0$,$y'(b)=0$

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

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

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

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

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

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

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

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

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