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588 Chapter 11 Boundary Value Problems and Fourier Expansions Proof Multiplying (11.2.2) by φn and integrating yields Z b a f(x)φn(x) dx = Z b a φn(x) ∞ X m=1 cmφm(x) ! dx. (11.2.4) It can be shown that the boundedness of the partial sums {fN}∞ N=1 and the integrability of f allow us to interchange the operations of in... | Elementary Differential Equations with Boundary Value Problems_Page_598_Chunk3001 |
Section 11.2 Fourier Expansions I 589 Fourier Series We’ll now study Fourier expansions in terms of the eigenfunctions 1, cos πx L , sin πx L , cos 2πx L , sin 2πx L , . . ., cos nπx L , sin nπx L , . . .. of Problem 5. If f is integrable on [−L, L], its Fourier expansion in terms of these functions is called the Fouri... | Elementary Differential Equations with Boundary Value Problems_Page_599_Chunk3002 |
590 Chapter 11 Boundary Value Problems and Fourier Expansions Theorem 11.2.4 If f is piecewise smooth on [−L, L], then the Fourier series F (x) = a0 + ∞ X n=1 an cos nπx L + bn sin nπx L (11.2.8) of f on [−L, L] converges for all x in [−L, L]; moreover, F (x) = f(x) if −L < x < L and f is cont... | Elementary Differential Equations with Boundary Value Problems_Page_600_Chunk3003 |
Section 11.2 Fourier Expansions I 591 1 2 1 2 −1 −2 x y Figure 11.2.1 Solution Note that wen’t bothered to define f(−2), f(0), and f(2). No matter how they may be defined, f is piecewise smooth on [−2, 2], and the coefficients in the Fourier series F (x) = a0 + ∞ X n=1 an cos nπx 2 + bn sin nπx 2 are not affected by t... | Elementary Differential Equations with Boundary Value Problems_Page_601_Chunk3004 |
592 Chapter 11 Boundary Value Problems and Fourier Expansions If n ≥1, then an = 1 2 Z 2 −2 f(x) cos nπx 2 dx = 1 2 Z 0 −2 (−x) cos nπx 2 dx + Z 2 0 1 2 cos nπx 2 dx = 2 n2π2 (cos nπ −1), and bn = 1 2 Z 2 −2 f(x) sin nπx 2 dx = 1 2 Z 0 −2 (−x) sin nπx 2 dx + Z 2 0 1 2 sin nπx 2 dx = 1 2nπ(1 + 3 cos nπ). Therefore... | Elementary Differential Equations with Boundary Value Problems_Page_602_Chunk3005 |
Section 11.2 Fourier Expansions I 593 Even and Odd Functions Computing the Fourier coefficients of a function f can be tedious; however, the computation can often be simplified by exploiting symmetries in f or some of its terms. To focus on this, we recall some concepts that you studied in calculus. Let u and v be defined... | Elementary Differential Equations with Boundary Value Problems_Page_603_Chunk3006 |
594 Chapter 11 Boundary Value Problems and Fourier Expansions We simplify the evaluation of these integrals by using Theorem 11.2.5 with u(x) = x2 and v(x) = x; thus, from (11.2.9), a0 = 1 2 Z 2 0 x2 dx = x3 6 2 0 = 4 3. From (11.2.10), an = Z 2 0 x2 cos nπx 2 dx = 2 nπ " x2 sin nπx 2 2 0 −2 Z 2 0 x sin nπx 2 dx # = 8 ... | Elementary Differential Equations with Boundary Value Problems_Page_604_Chunk3007 |
Section 11.2 Fourier Expansions I 595 1 −1 2 −2 1 2 3 4 5 6 x y Figure 11.2.3 Approximation of f(x) = x2 −x by partial sums of its Fourier series on [−2, 2] (b) If f is odd, the Fourier series of f on [−L, L] is F (x) = ∞ X n=1 bn sin nπx L , where bn = 2 L Z L 0 f(x) sin nπx L dx. Example 11.2.4 Find the Fourier serie... | Elementary Differential Equations with Boundary Value Problems_Page_605_Chunk3008 |
596 Chapter 11 Boundary Value Problems and Fourier Expansions 1 2 3 −1 −2 −3 1 2 3 −1 −2 −3 x y Figure 11.2.4 Approximation of f(x) = x by partial sums of its Fourier series on [−π, π] Theorem 11.2.4 implies that F (x) = 0, x = −π, x, −π < x < π, 0, x = π. Figure 11.2.4 shows how the partial sum Fm(x) = −2 m X n=... | Elementary Differential Equations with Boundary Value Problems_Page_606_Chunk3009 |
Section 11.2 Fourier Expansions I 597 and, if n ≥1, an = 2 π Z π 0 x cos nx dx = 2 nπ x sin nx π 0 − Z π 0 sin nx dx = 2 n2π cos nx π 0 = 2 n2π (cos nπ −1) = 2 n2π [(−1)n −1]. Therefore F (x) = π 2 + 2 π X n=0 (−1)n −1 n2 cos nx. (11.2.12) However, since (−1)n −1 = 0 if n = 2m, −2 if n = 2m + 1, the terms in (11.... | Elementary Differential Equations with Boundary Value Problems_Page_607_Chunk3010 |
598 Chapter 11 Boundary Value Problems and Fourier Expansions Example 11.2.7 (Gibbs Phenomenon) The Fourier series of f(x) = 0, −1 < x < −1 2, 1, −1 2 < x < 1 2, 0, 1 2 < x < 1 on [−1, 1] is F (x) = 1 2 + 2 π ∞ X n=1 (−1)n−1 2n −1 cos(2n −1)πx. (Verify.) According to Theorem 11.2.4, F (x) = ... | Elementary Differential Equations with Boundary Value Problems_Page_608_Chunk3011 |
Section 11.2 Fourier Expansions I 599 1 −1 y = 1.00 y = 1.09 y = − .09 x y Figure 11.2.7 The Gibbs Phenomenon: Example 11.2.7, N = 30 USING TECHNOLOGY The computation of Fourier coefficients will be tedious in many of the exercises in this chapter and the next. To learn the technique, we recommend that you do some exerc... | Elementary Differential Equations with Boundary Value Problems_Page_609_Chunk3012 |
600 Chapter 11 Boundary Value Problems and Fourier Expansions 6. C L = π; f(x) = x cos x 7. L = π; f(x) = |x| cosx 8. C L = π; f(x) = x sin x 9. L = π; f(x) = |x| sinx 10. L = 1; f(x) = 0, −1 < x < 1 2, cos πx, −1 2 < x < 1 2, 0, 1 2 < x < 1 11. L = 1; f(x) = 0, −1 < x < 1 2, x cos πx, −1 2 ... | Elementary Differential Equations with Boundary Value Problems_Page_610_Chunk3013 |
Section 11.2 Fourier Expansions I 601 21. Find the Fourier series of f(x) = (x −π) cos x on [−π, π]. 22. Find the Fourier series of f(x) = (x −π) sin x on [−π, π]. 23. Find the Fourier series of f(x) = sin kx (k ̸= integer) on [−π, π]. 24. Find the Fourier series of f(x) = cos kx (k ̸= integer) on [−π, π]. 25. (a) Supp... | Elementary Differential Equations with Boundary Value Problems_Page_611_Chunk3014 |
602 Chapter 11 Boundary Value Problems and Fourier Expansions and An = 2 L Z L 0 f(x) cos 2nπx L dx, Bn = 2 L Z L 0 f(x) sin 2nπx L dx, n = 1, 2, 3, . . .. L − L x y Figure 11.2.8 y = f(x), where f(x + L) = f(x), −L < x < 0 − L L x y Figure 11.2.9 y = f(x), where f(x + L) = −f(x), −L < x < 0 28. Show that if f is integ... | Elementary Differential Equations with Boundary Value Problems_Page_612_Chunk3015 |
Section 11.3 Fourier Expansions II 603 (a) Show that Z b a (f(x) −Fm(x))φn(x) dx = 0, n = 1, 2, . . ., m. (b) Show that Z b a (f(x) −Fm(x))2 dx ≤ Z b a (f(x) −Pm(x))2 dx, with equality if and only if an = cn, n = 1, 2, . . ., m. (c) Show that Z b a (f(x) −Fm(x))2 dx = Z b a f2(x) dx − m X n=1 c2 n Z b a φ2 n dx. (d) Co... | Elementary Differential Equations with Boundary Value Problems_Page_613_Chunk3016 |
604 Chapter 11 Boundary Value Problems and Fourier Expansions In this section we discuss Fourier expansions in terms of the eigenfunctions of Problems 1-4 for Sec- tion 11.1. Fourier Cosine Series From Exercise 11.1.20, the eigenfunctions 1, cos πx L , cos 2πx L , . . ., cos nπx L , . . . of the boundary value problem ... | Elementary Differential Equations with Boundary Value Problems_Page_614_Chunk3017 |
Section 11.3 Fourier Expansions II 605 L − L y = f(x) y = f(−x) x y Figure 11.3.1 Example 11.3.1 Find the Fourier cosine series of f(x) = x on [0, L]. Solution The coefficients are a0 = 1 L Z L 0 x dx = 1 L x2 2 L 0 = L 2 and, if n ≥1 an = 2 L Z L 0 x cos nπx L dx = 2 nπ " x sin nπx L L 0 − Z L 0 sin nπx L dx # = −2 nπ ... | Elementary Differential Equations with Boundary Value Problems_Page_615_Chunk3018 |
606 Chapter 11 Boundary Value Problems and Fourier Expansions From Exercise 11.1.19, the eigenfunctions sin πx L , sin 2πx L , . . ., sin nπx L , . . . of the boundary value problem y′′ + λy = 0, y(0) = 0, y(L) = 0 (Problem 1) are orthogonal on [0, L]. If f is integrable on [0, L] then the Fourier expansion of f in ter... | Elementary Differential Equations with Boundary Value Problems_Page_616_Chunk3019 |
Section 11.3 Fourier Expansions II 607 L − L y = f(x) y = − f(−x) x y Figure 11.3.2 Solution The coefficients are bn = 2 L Z L 0 x sin nπx L dx = −2 nπ " x cos nπx L L 0 − Z L 0 cos nπx L dx # = (−1)n+1 2L nπ + 2L n2π2 sin nπx L L 0 = (−1)n+1 2L nπ . Therefore S(x) = −2L π ∞ X n=1 (−1)n n sin nπx L . Theorem 11.3.2 impl... | Elementary Differential Equations with Boundary Value Problems_Page_617_Chunk3020 |
608 Chapter 11 Boundary Value Problems and Fourier Expansions (Problem 4) are orthogonal on [0, L]. If f is integrable on [0, L] then the Fourier expansion of f in terms of these functions is ∞ X n=1 cn cos (2n −1)πx 2L , where cn = Z L 0 f(x) cos (2n −1)πx 2L dx Z L 0 cos2 (2n −1)πx L dx = 2 L Z L 0 f(x) cos (2n −1)πx... | Elementary Differential Equations with Boundary Value Problems_Page_618_Chunk3021 |
Section 11.3 Fourier Expansions II 609 Theorem 11.3.3 If f is piecewise smooth on [0, L], then the mixed Fourier cosine series CM(x) = ∞ X n=1 cn cos (2n −1)πx 2L of f on [0, L], with cn = 2 L Z L 0 f(x) cos (2n −1)πx 2L dx, converges for all x in [0, L]; moreover, CM(x) = f(0+) if x = 0 f(x) ... | Elementary Differential Equations with Boundary Value Problems_Page_619_Chunk3022 |
610 Chapter 11 Boundary Value Problems and Fourier Expansions where dn = Z L 0 f(x) sin (2n −1)πx 2L dx Z L 0 sin2 (2n −1)πx 2L dx = 2 L Z L 0 f(x) sin (2n −1)πx 2L dx. We’ll call this expansion the mixed Fourier sine series of f on [0, L]. It can be shown (Exercise 58) that the mixed Fourier sine series of f on [0, L]... | Elementary Differential Equations with Boundary Value Problems_Page_620_Chunk3023 |
Section 11.3 Fourier Expansions II 611 converges for all x in [0, L]; moreover, SM (x) = 0 if x = 0 f(x) if 0 < x < L and f is continuous at x f(x−) + f(x+) 2 if 0 < x < L and f is discontinuous at x f(L−) if x = L. Example 11.3.4 Find the mixed Fourier sine series of f(x) = x on [0, L]. Solut... | Elementary Differential Equations with Boundary Value Problems_Page_621_Chunk3024 |
612 Chapter 11 Boundary Value Problems and Fourier Expansions with bn = −2L n2π2 Z L 0 f′′(x) sin nπx L dx. (11.3.5) (c) If f′(0) = f(L) = 0, then f(x) = ∞ X n=1 cn cos (2n −1)πx 2L , 0 ≤x ≤L, with cn = − 8L (2n −1)2π2 Z L 0 f′′(x) cos (2n −1)πx 2L dx. (11.3.6) (d) If f(0) = f′(L) = 0, then f(x) = ∞ X n=1 dn sin (2n −1... | Elementary Differential Equations with Boundary Value Problems_Page_622_Chunk3025 |
Section 11.3 Fourier Expansions II 613 Solution Here a0 = 1 L Z L 0 (3Lx2 −2x3) dx = 1 L Lx3 −x4 2 L 0 = L3 2 and an = 2 L Z L 0 (3Lx2 −2x3) cos nπx L dx, n ≥1. Evaluating this integral directly is laborious. However, since f′(x) = 6Lx −6x2, we see that f′(0) = f′(L) = 0. Since f′′′(x) = −12, we see from (11.3.4) t... | Elementary Differential Equations with Boundary Value Problems_Page_623_Chunk3026 |
614 Chapter 11 Boundary Value Problems and Fourier Expansions Therefore CM(x) = 32L3 π3 ∞ X n=1 1 (2n −1)3 (−1)n5 + 18 (2n −1)π cos (2n −1)πx 2L . Example 11.3.8 Find the mixed Fourier sine expansion of f(x) = x(2x2 −9Lx + 12L2) on [0, L]. Solution Since f(0) = f′(L) = 0, and f′′(x) = 6(2x −3L), we see from (11.3.7... | Elementary Differential Equations with Boundary Value Problems_Page_624_Chunk3027 |
Section 11.3 Fourier Expansions II 615 In Exercises 11-17 find the Fourier sine series. 11. C f(x) = 1; [0, L] 12. C f(x) = 1 −x; [0, 1] 13. f(x) = cos kx (k ̸= integer); [0, π] 14. C f(x) = 1, 0 ≤x ≤L 2 0, L 2 < x < L; [0, L] 15. C f(x) = x, 0 ≤x ≤L 2 , L −x, L 2 ≤x ≤L; [0, L]. 16. C f(x) = x sinx; [0, π] 17. f(x) ... | Elementary Differential Equations with Boundary Value Problems_Page_625_Chunk3028 |
616 Chapter 11 Boundary Value Problems and Fourier Expansions (b) In addition to the assumptions of Theorem 11.3.5(b), suppose f′′(0) = f′′(L) = 0, f′′′ is continuous, and f(4) is piecewise continuous on [0, L]. Show that bn = 2L3 n4π4 Z L 0 f(4)(x) sin nπx L dx, n ≥1. In Exercises 36-41 use Theorem 11.3.5(b) or, where... | Elementary Differential Equations with Boundary Value Problems_Page_626_Chunk3029 |
Section 11.3 Fourier Expansions II 617 53. f(x) = (x −L)3 + L3 54. f(x) = x(x2 −3L2) 55. f(x) = x3(3x −4L) 56. f(x) = x(x3 −2Lx2 + 2L3) 57. Show that the mixed Fourier cosine series of f on [0, L] is the restriction to [0, L] of the Fourier cosine series of f3(x) = f(x), 0 ≤x ≤L, −f(2L −x), L < x ≤2L on [0, 2L]. Use ... | Elementary Differential Equations with Boundary Value Problems_Page_627_Chunk3030 |
CHAPTER 12 Fourier Solutions of Partial Differential IN THIS CHAPTER we use the series discussed in Chapter 11 to solve partial differential equations that arise in problems of mathematical physics. SECTION 12.1 deals with the partial differential equation ut = a2uxx, which arises in problems of conduction of heat. SEC... | Elementary Differential Equations with Boundary Value Problems_Page_628_Chunk3031 |
Section 12.1 The Heat Equation 619 12.1 THE HEAT EQUATION We begin the study of partial differential equations with the problem of heat flow in a uniform bar of length L, situated on the x axis with one end at the origin and the other at x = L (Figure 12.1.1). We assume that the bar is perfectly insulated except possibl... | Elementary Differential Equations with Boundary Value Problems_Page_629_Chunk3032 |
620 Chapter 12 Fourier Solutions of Partial Differential for all (x, t). Since vt = XT ′ and vxx = X′′T, vt = a2vxx if and only if XT ′ = a2X′′T, which we rewrite as T ′ a2T = X′′ X . Since the expression on the left is independent of x while the one on the right is independent of t, this equation can hold for all (x, ... | Elementary Differential Equations with Boundary Value Problems_Page_630_Chunk3033 |
Section 12.1 The Heat Equation 621 Definition 12.1.1 The formal solution of the initial-boundary value problem ut = a2uxx, 0 < x < L, t > 0, u(0, t) = 0, u(L, t) = 0, t > 0, u(x, 0) = f(x), 0 ≤x ≤L (12.1.4) is u(x, t) = ∞ X n=1 αne−n2π2a2t/L2 sin nπx L , (12.1.5) where S(x) = ∞ X n=1 αn sin nπx L is the Fourier sine ser... | Elementary Differential Equations with Boundary Value Problems_Page_631_Chunk3034 |
622 Chapter 12 Fourier Solutions of Partial Differential Example 12.1.1 Solve (12.1.4) with f(x) = x(x2 −3Lx + 2L2). Solution From Example 11.3.6, the Fourier sine series of f on [0, L] is S(x) = 12L3 π3 ∞ X n=1 1 n3 sin nπx L . Therefore u(x, t) = 12L3 π3 ∞ X n=1 1 n3 e−n2π2a2t/L2 sin nπx L . If both ends of bar are i... | Elementary Differential Equations with Boundary Value Problems_Page_632_Chunk3035 |
Section 12.1 The Heat Equation 623 Definition 12.1.4 The formal solution of the initial-boundary value problem ut = a2uxx, 0 < x < L, t > 0, u(0, t) = 0, ux(L, t) = 0, t > 0, u(x, 0) = f(x), 0 ≤x ≤L (12.1.7) is u(x, t) = ∞ X n=1 αne−(2n−1)2π2a2t/4L2 sin (2n −1)πx 2L , where SM(x) = ∞ X n=1 αn sin (2n −1)πx 2L is the mix... | Elementary Differential Equations with Boundary Value Problems_Page_633_Chunk3036 |
624 Chapter 12 Fourier Solutions of Partial Differential L x u y = x Figure 12.1.2 Example 12.1.4 Solve (12.1.8) with f(x) = x −L. Solution From Example 11.3.3, the mixed Fourier cosine series of f on [0, L] is CM(x) = −8L π2 ∞ X n=1 1 (2n −1)2 cos (2n −1)πx 2L . Therefore u(x, t) = −8L π2 ∞ X n=1 1 (2n −1)2 e−(2n−1)2π... | Elementary Differential Equations with Boundary Value Problems_Page_634_Chunk3037 |
Section 12.1 The Heat Equation 625 so u satisfies (12.1.9) if v satisfies vt = a2vxx + a2q′′(x) + h(x), 0 < x < L, t > 0, v(0, t) = u0 −q(0), v(L, t) = uL −q(L), t > 0, v(x, 0) = f(x) −q(x), 0 ≤x ≤L. This reduces to vt = a2vxx, 0 < x < L, t > 0, v(0, t) = 0, v(L, t) = 0, t > 0, v(x, 0) = f(x) −q(x), 0 ≤x ≤L (12.1.11) if ... | Elementary Differential Equations with Boundary Value Problems_Page_635_Chunk3038 |
626 Chapter 12 Fourier Solutions of Partial Differential or u(0, t) = u0, ux(L, t) = uL, t > 0; however, this isn’t true in general for the boundary conditions ux(0, t) = u0, ux(L, t) = uL, t > 0. (See Exercise 47.) USING TECHNOLOGY Numerical experiments can enhance your understanding of the solutionsof initial-boundar... | Elementary Differential Equations with Boundary Value Problems_Page_636_Chunk3039 |
Section 12.1 The Heat Equation 627 12.1 Exercises 1. Explain Definition 12.1.3. 2. Explain Definition 12.1.4. 3. Explain Definition 12.1.5. 4. C Perform numerical experiments with the formal solution obtained in Example 12.1.1. 5. C Perform numerical experiments with the formal solution obtained in Example 12.1.2. 6. C Pe... | Elementary Differential Equations with Boundary Value Problems_Page_637_Chunk3040 |
628 Chapter 12 Fourier Solutions of Partial Differential 17. C ut = 9uxx, 0 < x < 4, t > 0, ux(0, t) = 0, ux(4, t) = 0, t > 0, u(x, 0) = x2, 0 ≤x ≤4 18. ut = 4uxx, 0 < x < 2, t > 0, ux(0, t) = 0, ux(2, t) = 0, t > 0, u(x, 0) = x(x −4), 0 ≤x ≤2 19. C ut = 9uxx, 0 < x < 1, t > 0, ux(0, t) = 0, ux(1, t) = 0, t > 0, u(x, 0... | Elementary Differential Equations with Boundary Value Problems_Page_638_Chunk3041 |
Section 12.1 The Heat Equation 629 32. ut = uxx, 0 < x < 1, t > 0, u(0, t) = 0, ux(1, t) = 0, t > 0, u(x, 0) = x(x3 −2x2 + 2), 0 ≤x ≤1 33. ut = 3uxx, 0 < x < π, t > 0, ux(0, t) = 0, u(π, t) = 0, t > 0, u(x, 0) = x2(π −x), 0 ≤x ≤π 34. ut = 16uxx, 0 < x < 2π, t > 0, ux(0, t) = 0, u(2π, t) = 0, t > 0, u(x, 0) = 4, 0 ≤x ≤2... | Elementary Differential Equations with Boundary Value Problems_Page_639_Chunk3042 |
630 Chapter 12 Fourier Solutions of Partial Differential 45. C ut = a2uxx, 0 < x < L, t > 0, ux(0, t) = 0, u(L, t) = 0, t > 0, u(x, 0) = 1, 0 ≤x ≤L 2 , 0, L 2 < x < L 46. C ut = a2uxx, 0 < x < L, t > 0, u(0, t) = 0, ux(L, t) = 0, t > 0, u(x, 0) = 1, 0 ≤x ≤L 2 , 0, L 2 < x < L 47. Let h be continuous on [0, L] and l... | Elementary Differential Equations with Boundary Value Problems_Page_640_Chunk3043 |
Section 12.2 The Wave Equation 631 (a) Show that u is defined for (x, t) such that t > 0. (b) For fixed t > 0, use Theorem 12.1.2 with z = x to show that ux(x, t) = π L ∞ X n=1 nαne−n2π2a2t/L2 cos nπx L , −∞< x < ∞. (c) Starting from the result of (a), use Theorem 12.1.2 with z = x to show that, for a fixed t > 0, uxx(x, ... | Elementary Differential Equations with Boundary Value Problems_Page_641_Chunk3044 |
632 Chapter 12 Fourier Solutions of Partial Differential Equations L x u Figure 12.2.1 A stretched string 1. The mass density (mass per unit length) ρ of the string is constant throughout the string. 2. The tension T induced by tightly stretching the string along the x-axis is so great that all other forces, such as gr... | Elementary Differential Equations with Boundary Value Problems_Page_642_Chunk3045 |
Section 12.2 The Wave Equation 633 T1 θ2 T2 x u θ1 Figure 12.2.2 A segment of the displaced string where ∆s is the length of the segment and x is the abscissa of the center of mass; hence, x < x < x + ∆x. From calculus, we know that ∆s = Z x+∆x x p 1 + u2x(σ, t) dσ; however, because of (12.2.2), we make the approximati... | Elementary Differential Equations with Boundary Value Problems_Page_643_Chunk3046 |
634 Chapter 12 Fourier Solutions of Partial Differential Equations Letting ∆x →0 yields uxx(x, t) = ρ T utt(x, t), which we rewrite as utt = a2uxx, with a2 = T/ρ. The Formal Solution As in Section 12.1, we use separation of variables to obtain a suitable definition for the formal solution of (12.2.1). We begin by lookin... | Elementary Differential Equations with Boundary Value Problems_Page_644_Chunk3047 |
Section 12.2 The Wave Equation 635 Then ∂vn ∂t (x, t) = −nπa L αn sin nπat L + βn cos nπat L sin nπx L , so vn(x, 0) = αn sin nπx L and ∂vn ∂t (x, 0) = βn sin nπx L . Therefore vn satisfies (12.2.1) with f(x) = αn sin nπx/L and g(x) = βn cos nπx/L. More generally, if α1, α2, ..., αm and β1, β2,..., βm are constants ... | Elementary Differential Equations with Boundary Value Problems_Page_645_Chunk3048 |
636 Chapter 12 Fourier Solutions of Partial Differential Equations yields cos nπat L sin nπx L = 1 2 sin nπ(x + at) L + sin nπ(x −at) L (12.2.10) and sin nπat L sin nπx L = −1 2 cos nπ(x + at) L −cos nπ(x −at) L = nπ 2L Z x+at x−at sin nπτ L dτ. (12.2.11) From (12.2.10), ∞ X n=1 αn cos nπat L sin nπx L = 1 2 ∞ ... | Elementary Differential Equations with Boundary Value Problems_Page_646_Chunk3049 |
Section 12.2 The Wave Equation 637 Proof Throughout this proof, k denotes an integer. Since f is differentiable on the open interval (0, L), both p and q are differentiable on every open interval ((k −1)L, kL). Thus, we need only to determine whether p and q are differentiable at x = kL for every k. (a) From Figure 12.... | Elementary Differential Equations with Boundary Value Problems_Page_647_Chunk3050 |
638 Chapter 12 Fourier Solutions of Partial Differential Equations Theorem 12.2.4 The formal solution of (12.2.1) is an actual solution if g is differentiable on [0, L] and g(0) = g(L) = 0, (12.2.15) while f is twice differentiable on [0, L] and f(0) = f(L) = 0 (12.2.16) and f′′ +(0) = f′′ −(L) = 0. (12.2.17) Proof We ... | Elementary Differential Equations with Boundary Value Problems_Page_648_Chunk3051 |
Section 12.2 The Wave Equation 639 Solution We leave it to you to verify that f and g satisfy the assumptions of Theorem 12.2.4. From Exercise 11.3.39, Sf(x) = 96L4 π5 ∞ X n=1 1 (2n −1)5 sin (2n −1)πx L . From Exercise 11.3.36, Sg(x) = 8L2 π3 ∞ X n=1 1 (2n −1)3 sin (2n −1)πx L . From (12.2.8), u(x, t) = 96L4 π5 ∞ X n=1... | Elementary Differential Equations with Boundary Value Problems_Page_649_Chunk3052 |
640 Chapter 12 Fourier Solutions of Partial Differential Equations L .5 L .5 L x y Figure 12.2.7 Graph of (12.2.21) which converges to f for all x in [0, L], by Theorem 11.3.2. Therefore u(x, t) = 4L π2 ∞ X n=1 (−1)n+1 (2n −1)2 cos (2n −1)πat L sin (2n −1)πx L . (12.2.22) This series converges absolutely for all (x, t)... | Elementary Differential Equations with Boundary Value Problems_Page_650_Chunk3053 |
Section 12.2 The Wave Equation 641 x = 0 x = L x = 2L x = − L x = − 2L x = − 3L x = 3L x y − .5L .5L Figure 12.2.8 The odd periodic extension of (12.2.21) x = 0 x = L x = 2L x = − L x = − 2L x = − 3L x = 3L x y .5L − .5L Figure 12.2.9 Graphs of y = Sf(x −at) (dashed) and y = Sf(x −at) (solid), with f as in (12.2.21) In... | Elementary Differential Equations with Boundary Value Problems_Page_651_Chunk3054 |
642 Chapter 12 Fourier Solutions of Partial Differential Equations L at .5L .5L x =.5L − at x =.5L + at x y Figure 12.2.10 The part of the graph from Figure 12.2.9 on [0, L] L .5L y = .5L − at x =.5L + at x = .5L − at x y Figure 12.2.11 The graph of (12.2.23) on [0, L] for a fixed t in (0, L/2a) USING TECHNOLOGY Althoug... | Elementary Differential Equations with Boundary Value Problems_Page_652_Chunk3055 |
Section 12.2 The Wave Equation 643 12.2 Exercises In Exercises 1-15 solve the initial-boundary value problem. In some of these exercises, Theorem 11.3.5(b) or Exercise 11.3.35 will simplify the computation of the coefficients in the Fourier sine series. 1. utt = 9uxx, 0 < x < 1, t > 0, u(0, t) = 0, u(1, t) = 0, t > 0, u... | Elementary Differential Equations with Boundary Value Problems_Page_653_Chunk3056 |
644 Chapter 12 Fourier Solutions of Partial Differential Equations 14. utt = 9uxx, 0 < x < 1, t > 0, u(0, t) = 0, u(1, t) = 0, t > 0, u(x, 0) = x(3x4 −10x2 + 7), ut(x, 0) = 0, 0 ≤x ≤1 15. C utt = 9uxx, 0 < x < 1, t > 0, u(0, t) = 0, u(1, t) = 0, t > 0, u(x, 0) = 0 ut(x, 0) = x(3x4 −10x2 + 7), 0 ≤x ≤1 16. We saw that th... | Elementary Differential Equations with Boundary Value Problems_Page_654_Chunk3057 |
Section 12.2 The Wave Equation 645 are the mixed Fourier cosine series of f and g on [0, L]; that is, αn = 2 L Z L 0 f(x) cos (2n −1)πx 2L dx and βn = 2 L Z L 0 g(x) cos (2n −1)πx 2L dx. In Exercises 18-31, use Exercise 17 to solve the initial-boundaryvalue problem. In some of these exercises Theorem 11.3.5(c) or Exerc... | Elementary Differential Equations with Boundary Value Problems_Page_655_Chunk3058 |
646 Chapter 12 Fourier Solutions of Partial Differential Equations 30. utt = uxx, 0 < x < 1, t > 0, ux(0, t) = 0, u(1, t) = 0, t > 0, u(x, 0) = x4 −4x3 + 6x2 −3, ut(x, 0) = 0, 0 ≤x ≤1 31. utt = uxx, 0 < x < 1, t > 0, ux(0, t) = 0, u(1, t) = 0, t > 0, u(x, 0) = 0, ut(x, 0) = x4 −4x3 + 6x2 −3, 0 ≤x ≤1 32. Adapt the proof... | Elementary Differential Equations with Boundary Value Problems_Page_656_Chunk3059 |
Section 12.2 The Wave Equation 647 38. C utt = 9uxx, 0 < x < 1, t > 0, u(0, t) = 0, ux(1, t) = 0, t > 0, u(x, 0) = 0, ut(x, 0) = x2(3 −2x), 0 ≤x ≤1 39. utt = 9uxx, 0 < x < 1, t > 0, u(0, t) = 0, ux(1, t) = 0, t > 0, u(x, 0) = (x −1)3 + 1, ut(x, 0) = 0, 0 ≤x ≤1 40. utt = 3uxx, 0 < x < π, t > 0, u(0, t) = 0, ux(π, t) = 0... | Elementary Differential Equations with Boundary Value Problems_Page_657_Chunk3060 |
648 Chapter 12 Fourier Solutions of Partial Differential Equations are the Fourier cosine series of f and g on [0, L]; that is, α0 = 1 L Z L 0 f(x) dx, β0 = 1 L Z L 0 g(x) dx, αn = 2 L Z L 0 f(x) cos nπx L dx, and βn = 2 L Z L 0 g(x) cos nπx L dx, n = 1, 2, 3, . . .. In Exercises 50-59 use Exercise 49 to solve the init... | Elementary Differential Equations with Boundary Value Problems_Page_658_Chunk3061 |
Section 12.2 The Wave Equation 649 62. Suppose λ and µ are constants and either pn(x) = cos nλx or pn(x) = sin nλx, while either qn(t) = cos nµt or qn(t) = sin nµt for n = 1, 2, 3, .... Let u(x, t) = ∞ X n=1 knpn(x)qn(t), (A) where {kn}∞ n=1 are constants. (a) Show that if P∞ n=1 |kn| converges then u(x, t) converges f... | Elementary Differential Equations with Boundary Value Problems_Page_659_Chunk3062 |
650 Chapter 12 Fourier Solutions of Partial Differential Equations (c) Solve utt = a2uxx, −∞< t < ∞, t > 0, u(x, 0) = f(x), ut(x, 0) = g(x), −∞< x < ∞. In Exercises 64-68 use the result of Exercise 63 to find a solution of utt = a2uxx, −∞< x < ∞ that satisfies the given initial conditions. 64. u(x, 0) = x, ut(x, 0) = 4ax... | Elementary Differential Equations with Boundary Value Problems_Page_660_Chunk3063 |
Section 12.3 Laplace’s Equation in Rectangular Coordinates 651 y x a b Figure 12.3.1 A rectangular region and its boundary where α, β, γ, and δ can each be either 0 or 1; thus, there are 16 possibilities. Let BVP(α, β, γ, δ)(f0, f1, g0, g1) denote the problem of finding a solution of (12.3.2) that satisfies these conditi... | Elementary Differential Equations with Boundary Value Problems_Page_661_Chunk3064 |
652 Chapter 12 Fourier Solutions of Partial Differential Equations BVP(α, β, γ, δ)(0, 0, g0, 0), and BVP(α, β, γ, δ)(0, 0, 0, g1) is a solution of BVP(α, β, γ, δ)(f0, f1, g0, g1). Therefore we concentrate on problems where only one of the functions f0, f1, g0, g2 isn’t identically zero. There are 64 (count them!) probl... | Elementary Differential Equations with Boundary Value Problems_Page_662_Chunk3065 |
Section 12.3 Laplace’s Equation in Rectangular Coordinates 653 y x a b uxx + uyy = 0 u(x,0) = f(x) u(x,b) = 0 u(0,y) = 0 u(a,y) = 0 Figure 12.3.4 The boundary value problem (12.3.4) however, because of the nonhomogeneous Dirichlet condition at y = 0, it’s better to require that Yn(0) = 1, which can be achieved by divid... | Elementary Differential Equations with Boundary Value Problems_Page_663_Chunk3066 |
654 Chapter 12 Fourier Solutions of Partial Differential Equations is the Fourier sine series of f on [0, a]; that is, αn = 2 a Z a 0 f(x) sin nπx a dx, n = 1, 2, 3, . . .. If y < b then sinh nπ(b −y)/a sinh nπb/a ≈e−nπy/a (12.3.9) for large n, so the series in (12.3.8) converges if 0 < y < b; moreover, since also cosh... | Elementary Differential Equations with Boundary Value Problems_Page_664_Chunk3067 |
Section 12.3 Laplace’s Equation in Rectangular Coordinates 655 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 Figure 12.3.5 1 2 1 2 3 x y Figure 12.3.6 Example 12.3.3 Define the formal solution of uxx + uyy = 0, 0 < x < a, 0 < y < b, u(x, 0) = 0, uy(x, b) = f(x), 0 ≤x ≤a, ux(0, y) = 0, ux(... | Elementary Differential Equations with Boundary Value Problems_Page_665_Chunk3068 |
656 Chapter 12 Fourier Solutions of Partial Differential Equations X′(a) = Y (0) = 0; hence, we let k = −λ in (12.3.3). Thus, X and Y must satisfy X′′ + λX = 0, X′(0) = 0, X′(a) = 0 (12.3.12) and Y ′′ −λY = 0, Y (0) = 0. (12.3.13) From Theorem 11.1.3, the eigenvalues of (12.3.12) are λ = 0, with associated eigenfunctio... | Elementary Differential Equations with Boundary Value Problems_Page_666_Chunk3069 |
Section 12.3 Laplace’s Equation in Rectangular Coordinates 657 Example 12.3.4 Solve (12.3.11) with f(x) = x. Solution From Example 11.3.1, C(x) = a 2 −4a π2 ∞ X n=1 1 (2n −1)2 cos (2n −1)πx a . Therefore u(x, y) = ay 2 −4a2 π3 ∞ X n=1 sinh(2n −1)πy/a (2n −1)3 cosh(2n −1)πb/a cos (2n −1)πx a . (12.3.15) For graphing pur... | Elementary Differential Equations with Boundary Value Problems_Page_667_Chunk3070 |
658 Chapter 12 Fourier Solutions of Partial Differential Equations y x a b uxx + uyy = 0 u(x,0) = 0 uy(x,b) = 0 u(0,y) = g(y) ux(a,y) = 0 Figure 12.3.10 The boundary value problem (12.3.16) and Y ′′ + λY = 0, Y (0) = 0, Y ′(b) = 0. (12.3.18) From Theorem 11.1.4, the eigenvalues of (12.3.18) are λn = (2n −1)2π2/4b2, wit... | Elementary Differential Equations with Boundary Value Problems_Page_668_Chunk3071 |
Section 12.3 Laplace’s Equation in Rectangular Coordinates 659 Therefore vn satisfies (12.3.16) with g(y) = sin(2n −1)πy/2b. More generally, if α1, ..., αm are arbitrary constants then um(x, y) = m X n=1 αn cosh(2n −1)π(x −a)/2b cosh(2n −1)πa/2b sin (2n −1)πy 2b satisfies (12.3.16) with g(y) = m X n=1 αn sin (2n −1)πy 2b... | Elementary Differential Equations with Boundary Value Problems_Page_669_Chunk3072 |
660 Chapter 12 Fourier Solutions of Partial Differential Equations y x a b uxx + uyy = 0 uy(x,0) = 0 u(x,b) = 0 ux(0,y) = 0 ux(a,y) = g(y) Figure 12.3.11 The boundary value problem (12.3.20) From Theorem 11.1.4, the eigenvalues of (12.3.22) are λn = (2n −1)2π2/4b2, with associated eigen- functions Yn = cos (2n −1)πy 2b... | Elementary Differential Equations with Boundary Value Problems_Page_670_Chunk3073 |
Section 12.3 Laplace’s Equation in Rectangular Coordinates 661 satisfies (12.3.20) with g(y) = ∞ X n=1 αn cos (2n −1)πy 2b . Therefore, if g is an arbitrary piecewise smooth function on [0, b], we define the formal solution of (12.3.20) to be u(x, y) = 2b π ∞ X n=1 αn cosh(2n −1)πx/2b (2n −1) sinh(2n −1)πa/2b cos (2n −1)... | Elementary Differential Equations with Boundary Value Problems_Page_671_Chunk3074 |
662 Chapter 12 Fourier Solutions of Partial Differential Equations y x a uxx + uyy = 0 u(x,0) = f(x) u(0,y) = 0 u(a,y) = 0 Figure 12.3.12 A boundary value problem on a semi-infinite strip Example 12.3.9 Define the bounded formal solution of (12.3.24). Solution Proceeding as in the solution of Example 12.3.1, we find that ... | Elementary Differential Equations with Boundary Value Problems_Page_672_Chunk3075 |
Section 12.3 Laplace’s Equation in Rectangular Coordinates 663 See Exercises 29-34 for other boundary value problems on a semi-infinite strip. 12.3 Exercises In Exercises 1-16 apply the definition developed in Example 1 to solve the boundary value problem. (Use Theorem 11.3.5 where it applies.) Where indicated by C , gra... | Elementary Differential Equations with Boundary Value Problems_Page_673_Chunk3076 |
664 Chapter 12 Fourier Solutions of Partial Differential Equations 13. C uxx + uyy = 0, 0 < x < 1, 0 < y < π, uy(x, 0) = 0, u(x, π) = 0, 0 ≤x ≤1, ux(0, y) = 0, ux(1, y) = sin y, 0 ≤y ≤π 14. uxx + uyy = 0, 0 < x < 2, 0 < y < 3, uy(x, 0) = 0, u(x, 3) = 0, 0 ≤x ≤2, ux(0, y) = 0, ux(2, y) = y(3 −y), 0 ≤y ≤3 15. uxx + uyy =... | Elementary Differential Equations with Boundary Value Problems_Page_674_Chunk3077 |
Section 12.3 Laplace’s Equation in Rectangular Coordinates 665 25. C uy(x, 0) = 0, u(x, b) = 0, 0 < x < a, ux(0, y) = 0, u(a, y) = g(y), 0 < y < b a = 2, b = 2, g(y) = 4 −y2 26. u(x, 0) = 0, u(x, b) = 0, 0 < x < a, ux(0, y) = 0, ux(a, y) = g(y), 0 < y < b a = 1, b = 4, g(y) = y, 0 ≤y ≤2, 4 −y, 2 ≤y ≤4 27. u(x, 0) = 0... | Elementary Differential Equations with Boundary Value Problems_Page_675_Chunk3078 |
666 Chapter 12 Fourier Solutions of Partial Differential Equations has no solution unless Z a 0 f0(x) dx = Z a 0 f1(x) dx = Z b 0 g0(y) dy = Z b 0 g1(y) dy = 0. In this case it has infinitely many formal solutions. Find them. 37. In this exercise take it as given that the infinite series P∞ n=1 npe−qn converges for all p... | Elementary Differential Equations with Boundary Value Problems_Page_676_Chunk3079 |
Section 12.4 Laplace’s Equation in Polar Coordinates 667 (g) Conclude that u satisfies Laplace’s equation for all (x, y) such that 0 < y < b. By repeatedly applying the arguments in (c)–(f), it can be shown that u can be differentiated term by term any number of times with respect to x and/or y if 0 < y < b. 12.4 LAPLAC... | Elementary Differential Equations with Boundary Value Problems_Page_677_Chunk3080 |
668 Chapter 12 Fourier Solutions of Partial Differential Equations We first look for products v(r, θ) = R(r)Θ(θ) that satisfy (12.4.1). For this function, vrr + 1 r vr + 1 r2 vθθ = R′′Θ + 1 r R′Θ + 1 r2 RΘ′′ = 0 for all (r, θ) with r ̸= 0 if r2R′′ + rR′ R = −Θ′′ Θ = λ, where λ is a separation constant. (Verify.) This eq... | Elementary Differential Equations with Boundary Value Problems_Page_678_Chunk3081 |
Section 12.4 Laplace’s Equation in Polar Coordinates 669 so the general solution of (12.4.6) is Rn = c1rn + c2r−n, (12.4.7) by Theorem 7.4.3. Consistent with our previous assumption on R0, we now require Rn to be bounded as r →0+. This implies that c2 = 0, and we choose c1 = ρ−n. Then Rn(r) = rn/ρn, so vn(r, θ) = Rn(r)... | Elementary Differential Equations with Boundary Value Problems_Page_679_Chunk3082 |
670 Chapter 12 Fourier Solutions of Partial Differential Equations Solution From Example 11.2.6, θ(π2 −θ2) = 12 ∞ X n=1 (−1)n n3 sin nθ, −π ≤θ ≤π, so u(r, θ) = 12 ∞ X n=1 rn ρn (−1)n n3 sin nθ, 0 ≤r ≤ρ, −π ≤θ ≤π. Example 12.4.2 Define the formal solution of urr + 1 r ur + 1 r2 uθθ = 0, ρ0 < r < ρ, −π ≤θ < π, u(ρ0, θ) = ... | Elementary Differential Equations with Boundary Value Problems_Page_680_Chunk3083 |
Section 12.4 Laplace’s Equation in Polar Coordinates 671 satisfy these requirements. Therefore v0(ρ, θ) = lnr/ρ0 ln ρ/ρ0 and vn(r, θ) = ρ−n 0 rn −ρn 0 r−n ρ−n 0 ρn −ρn 0 ρ−n (αn cos nθ + βn sin nθ), n = 1, 2, 3, . . ., where αn and βn are arbitrary constants. If α0, α1,..., αm and β1, β2, ..., βm are arbitrary constant... | Elementary Differential Equations with Boundary Value Problems_Page_681_Chunk3084 |
672 Chapter 12 Fourier Solutions of Partial Differential Equations ur r + r−1 ur + r−2 uθ θ = 0 u(ρ,θ) = f(θ) u(r,γ) = 0 γ x y Figure 12.4.3 The boundary value problem (12.4.10) The indicial polynomial of this equation is s(s −1) + s −n2π2 γ2 = s −nπ γ s + nπ γ , so Rn = c1rnπ/γ + c2r−nπ/γ, by Theorem 7.4.3. To... | Elementary Differential Equations with Boundary Value Problems_Page_682_Chunk3085 |
Section 12.4 Laplace’s Equation in Polar Coordinates 673 This motivates us to define the bounded formal solution of (12.4.10) to be um(r, θ) = ∞ X n=1 αn rnπ/γ ρnπ/γ sin nπθ γ , where S(θ) = ∞ X n=1 αn sin nπθ γ is the Fourier sine expansion of f on [0, γ]; that is, αn = 2 γ Z γ 0 f(θ) sin nπθ γ dθ. 12.4 Exercises 1. De... | Elementary Differential Equations with Boundary Value Problems_Page_683_Chunk3086 |
674 Chapter 12 Fourier Solutions of Partial Differential Equations 5. Define the formal solution of urr + 1 r ur + 1 r2 uθθ = 0, ρ0 < r < ρ, 0 < θ < γ, ur(ρ0, θ) = g(θ), ur(ρ, θ) = 0, 0 ≤θ ≤γ, u(r, 0) = 0, uθ(r, γ) = 0, ρ0 < r < ρ, where 0 < γ < 2π and 0 < ρ0 < ρ. 6. Define the bounded formal solution of urr + 1 r ur + 1... | Elementary Differential Equations with Boundary Value Problems_Page_684_Chunk3087 |
CHAPTER 13 Boundary Value Problems for Second Order Ordinary Differential Equations IN THIS CHAPTER we discuss boundary value problems and eigenvalue problems for linear second order ordinary differential equations. Section 13.1 discusses point two-point boundary value problems for linear second order ordinary differ- ... | Elementary Differential Equations with Boundary Value Problems_Page_686_Chunk3088 |
Section 13.1 Two-Point Boundary Value Problems 677 13.1 TWO-POINT BOUNDARY VALUE PROBLEMS In Section 5.3 we considered initial value problems for the linear second order equation P0(x)y′′ + P1(x)y′ + P2(x)y = F (x). (13.1.1) Suppose P0, P1, P2, and F are continuous and P0 has no zeros on an open interval (a, b). From T... | Elementary Differential Equations with Boundary Value Problems_Page_687_Chunk3089 |
678 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations This boundary value problem is homogeneous if F = 0 and k1 = k2 = 0; otherwise it’s nonhomoge- neous. We leave it to you (Exercise 1) to verify that B1 and B2 are linear operators; that is, if c1 and c2 are constants then Bi(c1y1 + ... | Elementary Differential Equations with Boundary Value Problems_Page_688_Chunk3090 |
Section 13.1 Two-Point Boundary Value Problems 679 Proof Recall that B1(z) = αz(a) + βz′(a) and α2 + β2 ̸= 0. Therefore, if B1(z1) = B1(z2) = 0 then (α, β) is a nontrivial solution of the system αz1(a) + βz′ 1(a) = 0 αz2(a) + βz′(a) = 0. This implies that z1(a)z′ 2(a) −z′ 1(a)z2(a) = 0, so {z1, z2} is linearly dependen... | Elementary Differential Equations with Boundary Value Problems_Page_689_Chunk3091 |
680 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations Since the determinant of a product of matrices is the product of the determinants of the matrices, (13.1.8) and (13.1.12) imply (13.1.9). (b) =⇒ (c): Since {y1, y2} is a fundamental set of solutions of Ly = 0, the general solution o... | Elementary Differential Equations with Boundary Value Problems_Page_690_Chunk3092 |
Section 13.1 Two-Point Boundary Value Problems 681 Therefore the solution of (13.1.13) can be written as y = x3 + c1x + c2x2. Then y′ = 3x2 + c1 + 2c2x, and imposing the boundary conditions yields the system c1 + c2 = 3 c1 + 4c2 = −9, so c1 = 7 and c2 = −4. Therefore y = x3 + 7x −4x2 is the unique solution of (13.1.13)... | Elementary Differential Equations with Boundary Value Problems_Page_691_Chunk3093 |
682 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations Theorem 13.1.3 Suppose the homogeneous boundary value problem Ly = 0, B1(y) = 0, B2(y) = 0 (13.1.15) has only the trivial solution. Let y1 and y2 be linearly independent solutions of Ly = 0 such that B1(y1) = 0 and B2(y2) = 0, and l... | Elementary Differential Equations with Boundary Value Problems_Page_692_Chunk3094 |
Section 13.1 Two-Point Boundary Value Problems 683 This is the Green’s function for the boundary value problem (13.1.16). The Green’s function is related to the boundary value problem (13.1.16) in much the same way that the inverse of a square matrix A is related to the linear algebraic system y = Ax; just as we substi... | Elementary Differential Equations with Boundary Value Problems_Page_693_Chunk3095 |
684 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations The Green’s function is G(x, t) = (cos t −sin t)(cos x + sin x) 2 , 0 ≤t ≤x, (cos x −sin x)(cos t + sin t) 2 , x ≤t ≤π. We’ll now consider the situation not covered by Theorem 13.1.3. Theorem 13.1.4 Suppose the homogen... | Elementary Differential Equations with Boundary Value Problems_Page_694_Chunk3096 |
Section 13.1 Two-Point Boundary Value Problems 685 Example 13.1.7 Applying Theorem 13.1.4 to the boundary value problem y′′ + y = F (x), y(0) = 0, y(π) = 0 (13.1.25) explains the Examples 13.1.2 and 13.1.3. The complementary equation y′′ + y = 0 has the linear inde- pendent solutions y1 = sin x and y2 = cos x, and y1 s... | Elementary Differential Equations with Boundary Value Problems_Page_695_Chunk3097 |
686 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations 9. (a) State a condition on a and b such that the boundary value problem y′′ + y = F (x), y(a) = 0, y(b) = 0 (A) has a unique solution for every continuous F , and find the solution by the method used to prove Theorem 13.1.3 (b) In t... | Elementary Differential Equations with Boundary Value Problems_Page_696_Chunk3098 |
Section 13.1 Two-Point Boundary Value Problems 687 22. Find the Green’s function for the boundary value problem y′′ = F (x), y(0) −2y′(0) = 0, y(1) + 2y′(1) = 0. (A) Then use the Green’s function to solve (A) with (a) F (x) = 1, (b) F (x) = x, and (c) F (x) = x2. 23. Find the Green’s function for the boundary value pro... | Elementary Differential Equations with Boundary Value Problems_Page_697_Chunk3099 |
688 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations 13.2 STURM-LIOUVILLE PROBLEMS In this section we consider eigenvalue problems of the form P0(x)y′′ + P1(x)y′ + P2(x)y + λR(x)y = 0, B1(y) = 0, B2(y) = 0, (13.2.1) where B1(y) = αy(a) + βy′(a) and B2(y) = ρy(b) + δy′(b). As in Sectio... | Elementary Differential Equations with Boundary Value Problems_Page_698_Chunk3100 |
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