13.2E: Sturm-Liouville Problems (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
Q13.2.1
In Exercises 13.2.1-13.2.7 rewrite the equation in Sturm-Liouville form (with λ=0). Assume that b,c,α, and ν are constants.
1. y″+by′+cy=0
2. x2y″+xy′+(x2−ν2)y=0 (Bessel’s equation)
3. (1−x2)y″−xy′+α2y=0 (Chebyshev’s equation)
4. x2y″+bxy′+cy=0 (Euler’s equation)
5. y″−2xy′+2αy=0 (Hermite’s equation)
6. xy″+(1−x)y′+αy=0 (Laguerre’s equation)
7. (1−x2)y″−2xy′+α(α+1)y=0 (Legendre’s equation)
Q13.2.2
8. In Example 13.2.4 we found that the eigenvalue problem x2y″+xy′+λy=0,y(1)=0,y(2)=0is equivalent to the Sturm-Liouville problem (xy′)′+λxy=0,y(1)=0,y(2)=0. Multiply the differential equation in (B) by y and integrate to show that λ∫21y2(x)xdx=∫21x(y′(x))2dx. Conclude from this that the eigenvalues of (A) are all positive.
9. Solve the eigenvalue problem y″+2y′+y+λy=0,y(0)=0,y(1)=0.
10. Solve the eigenvalue problem y″+2y′+y+λy=0,y′(0)=0,y′(1)=0.
Q13.2.3
In Exercises 13.2.11-13.2.20:
- Determine whether λ=0 is an eigenvalue. If it is, find an associated eigenfunction.
- Compute the negative eigenvalues with errors not greater than 5×10−8. State the form of the associated eigenfunctions.
- Compute the first four positive eigenvalues with errors not greater than 5×10−8. State the form of the associated eigenfunctions.
11. y″+λy=0,y(0)+2y′(0)=0,y(2)=0
12. y″+λy=0,y′(0)=0,y(1)−2y′(1)=0
13. y″+λy=0,y(0)−y′(0)=0,y′(π)=0
14. y″+λy=0,y(0)+2y′(0)=0,y(π)=0
15. y″+λy=0,y′(0)=0,y(2)−y′(2)=0
16. y″+λy=0,y(0)+y′(0)=0,y(2)+2y′(2)=0
17. y″+λy=0,y(0)+2y′(0)=0,y(3)−2y′(3)=0
18. y″+λy=0,3y(0)+y′(0)=0,3y(2)−2y′(2)=0
19. y″+λy=0,y(0)+2y′(0)=0,y(3)−y′(3)=0
20. y″+λy=0,5y(0)+2y′(0)=0,5y(1)−2y′(1)=0
Q13.2.4
21. Find the first five eigenvalues of the boundary value problem
y″+2y′+y+λy=0,y(0)=0,y′(1)=0
with errors not greater than 5×10−8. State the form of the associated eigenfunctions.
Q13.2.5
In Exercises 13.2.22-13.2.24 take it as given that {xekx,xe−kx} and {xcoskx,xsinkx} are fundamental sets of solutions of x2y″−2xy′+2y−k2x2y=0 and x2y″−2xy′+2y+k2x2y=0 respectively.
22. Solve the eigenvalue problem for
x2y″−2xy′+2y+λx2y=0,y(1)=0,y(2)=0.
23. Find the first five eigenvalues of
x2y″−2xy′+2y+λx2y=0,y′(1)=0,y(2)=0
with errors no greater than 5×10−8. State the form of the associated eienfunctions.
24. Find the first five eigenvalues of
x2y″−2xy′+2y+λx2y=0,y(1)=0,y′(2)=0
with errors no greater than 5×10−8. State the form of the associated eienfunctions.
Q13.2.6
25. Consider the Sturm-Liouville problem
y″+λy=0,y(0)=0,y(L)+δy′(L)=0.
- Show that (A) can’t have more than one negative eigenvalue, and find the values of δ for which it has one.
- Find all values of δ such that λ=0 is an eigenvalue of (A).
- Show that λ=k2 with k>0 is an eigenvalue of (A) if and only if tankL=−δk.
- For n=1, 2, …, let yn be an eigenfunction associated with λn=k2n. From Theorem 13.2.4, ym and yn are orthogonal over [0,L] if m≠n. Verify this directly. (B).
26. Solve the Sturm-Liouville problem
y″+λy=0,y(0)+αy′(0)=0,y(π)+αy′(π)=0,
where α≠0.
27. Consider the Sturm-Liouville problem
y″+λy=0,y(0)+αy′(0)=0,y(1)+(α−1)y′(1)=0,
where 0<α<1.
- Show that λ=0 is an eigenvalue of (A), and find an associated eigenfunction.
- Show that (A) has a negative eigenvalue, and find the form of an associated eigenfunction.
- Give a graphical argument to show that (A) has infinitely many positive eigenvalues λ1<λ2<⋯<λn<⋯, and state the form of the associated eigenfunctions.
Q13.2.7
Exercises 13.2.28-13.2.30 deal with the Sturm-Liouville problem y″+λy=0,αy(0)+βy′(0),ρy(L)+δy′(L)=0, where α2+β2>0 and ρ2+δ2>0.
28. Show that λ=0 is an eigenvalue of (SL) if and only if α(ρL+δ)−βρ=0.
29. The point of this exercise is that (SL) can’t have more than two negative eigenvalues.
- Show that λ is a negative eigenvalue of (SL) if and only if λ=−k2, where k is a positive solution of (αρ−βδk2)sinhkL+k(αδ−βρ)coshkL.
- Suppose αδ−βρ=0. Show that (SL) has a negative eigenvalue if and only if αρ and βδ are both nonzero. Find the negative eigenvalue and an associated eigenfunction. HINT: Show that in this case ρ=pα and s=qβ, where q≠0.
- Suppose βρ−αδ≠0. We know from Section 11.1 that (SL) has no negative eigenvalues if αρ=0 and βδ=0. Assume that either αρ≠0 or βδ≠0. Then we can rewrite (A) as tanhkL=k(βρ−αδ)αρ−βδk2. By graphing both sides of this equation on the same axes (there are several possibilities for the right side), show that it has at most two positive solutions, so (SL) has at most two negative eigenvalues.
30. The point of this exercise is that (SL) has infinitely many positive eigenvalues λ1<λ2<⋯<λn<⋯, and that lim.
- Show that \lambda is a positive eigenvalue of (SL) if and only if \lambda=k^{2}, where k is a positive solution of (\alpha\rho+\beta\delta k^{2})\sin kL+k(\alpha\delta-\beta\rho)\cos kL=0.\tag{A}
- Suppose \alpha\delta-\beta\rho=0. Show that the positive eigenvalues of (SL) are \lambda_{n}=(n\pi/L)^{2}, n=1, 2, 3, …. HINT: Recall the hint in Exercise 13.2.29b.
Now suppose \alpha\delta-\beta\rho\ne0. From Section 11.1, if \alpha\rho=0 and \beta\delta=0, then (SL) has the eigenvalues \lambda_{n}=[(2n-1)\pi/2L]^{2},\quad n=1,2,3, \dots\nonumber (why?), so let’s suppose in addition that at least one of the products \alpha\rho and \beta\delta is nonzero. Then we can rewrite (A) as \tan kL= \frac{k(\beta\rho-\alpha\delta)} {\alpha\rho-\beta\delta k^{2}}.\tag{B} By graphing both sides of this equation on the same axes (there are several possibilities for the right side), convince yourself of the following: - If \beta\delta=0, there’s a positive integer N such that (B) has one solution k_{n} in each of the intervals \left((2n-1)\pi/L, (2n+1)\pi/L)\right),\quad n=N,N+1,N+2,\dots,\tag{C} and either \lim_{n\to\infty}\left(k_{n}-\frac{(2n-1)\pi}{2L}\right) =0\quad \text{or} \quad \lim_{n\to\infty}\left(k_{n}-\frac{(2n+1)\pi}{2L}\right)=0.\nonumber
- If \beta\delta\ne0, there’s a positive integer N such that (B) has one solution k_{n} in each of the intervals (C) and \lim_{n\to\infty}\left(k_{n}-\frac{n\pi}{N}\right)=0.\nonumber
31. The following Sturm–Liouville problems are generalizations of Problems 1–4 of Section 11.1.
Problem 1: (p(x)y')'+\lambda r(x)y=0, y(a)=0,y(b)=0
Problem 2: (p(x)y')'+\lambda r(x)y=0, y'(a)=0,y'(b)=0
Problem 3: (p(x)y')'+\lambda r(x)y=0, y(a)=0,y'(b)=0
Problem 4: (p(x)y')'+\lambda r(x)y=0, y'(a)=0,y(b)=0
Prove: Problems 1–4 have no negative eigenvalues. Moreover, \lambda=0 is an eigenvalue of Problem 2 with associated eigenfunction y_{0}=1, but \lambda=0 isn’t an eigenvalue of Problems 1, 3, and 4. HINT: See the proof of Theorem 11.1.1.
32. Show that the eigenvalues of the Sturm–Liouville problem
(p(x)y')'+\lambda r(x)y=0,\quad \alpha y(a)+ \beta y'(a)=0,\quad \rho y(b)+\delta y'(b)\nonumber
are all positive if \alpha\beta\le 0, \rho\delta\ge 0, and (\alpha\beta)^{2}+(\rho\delta)^{2}>0.