Math 412-200 — Summer II, 2011

Assignment 1.

• Read sections 1.1-1.5, 2.1-2.3, 2.4.2.
• Do exercises 1.2.3, 1.3.2, 1.4.1(f), 1.4.3, 2.3.1(a,b), 2.3.2(d,e), 2.3.3(a,b), 2.3.6, 2.4.2 [Hint: Separete variables and use your solution to 2.3.2(e).]

Assignment 2.

• Read sections 2.5, 3.1-3.3.
• Do exercises 2.4.1 (see §2.4.1), 2.5.3(a), 2.5.5(a), 2.5.8(a,b), 2.5.10, 2.5.19, 3.2.1(c,e,f), 3.2.2(b,f).
• Solve the problem of heat flow in a ring (see § 2.4.2), given that the initial temperature is

  
  where T is a constant.

Assignment 3.

• Read sections 3.4, 3.5, 4.4, 12.3
• Do exercises 3.3.1(b,d), 3.3.2(a,d), 3.3.5(a), 3.3.15, 3.3.18. (Hint: in 3.3.18, look at the appropriate periodic extension for each case.), 3.4.5, 3.5.2, 3.5.7
• Differentiate the cosine series obtained in 3.3.5(a) to find the sine series for f(x) = x, where 0 ≤ x ≤ L.
• Integrate the cosine series in 3.3.5(a) to find the sine series for x³ − L²x.

Assignment 4.

• Read sections 5.1-5.5
• Do exercises 4.4.3(b), 4.4.8, 4.4.9, 4.4.10(a,b), 4.4.13(a), 12.3.1, 12.3.5, 5.3.4(a,b), 5.3.5, 5.3.8, 5.3.10(a,b,c,d,e).

Assignment 5.

• Read sections 5.6-5.8, 5.10, 7.1-7.3
• Do exercises 5.4.2, 5.5.1(a,c), 5.5.3, 5.6.1(b), 5.7.2, 5.8.3, 5.8.8(a,b,c), 5.10.2(b), 5.10.4(a).

• For each of the Sturm-Liouville problems below, show, by integrating by parts, that the eigenfunctions corresponding to distinct eigenvalues are orthogonal.
  1. p(x) = (x²+1)⁻¹, σ = 2, q(x) = −x; φ(0) = 0 and φ(1) = 0.
  2. p(x) = 1, σ = x; q = 0; φ(0) = 0 and φ(2) + φ′(2) = 0.
  3. p(x) = 1, σ = 1; q = 0; φ(0) = φ(1) and φ′(0) = φ′(1).

Assignment 6.

• Read sectionns 7.7, 7.8
• Do exercises 7.7.1, 7.7.2(d), 7.7.3(a), 7.7.7, 7.7.11, 7.12(d,e)

Assignment 7.

• Read sectionns 10.1-10.3, 10.4.2, 10.4.3
• Do exercises 10.3.1, 10.3.5, 10.3.7, 10.3.8, 10.4.3(a)