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Section 13.2 Sturm-Liouville Problems 689 In this case the general solution of the differential equation in (13.2.2) is y = e−3x/2(c1 cos ωx + c2 sinωx). The boundary condition y(0) = 0 requires that c1 = 0, so y = c2e−3x/2 sin ωx, which holds with c2 ̸= 0 if and only if ω = nπ, where n is an integer. We may assume tha... | Elementary Differential Equations with Boundary Value Problems_Page_699_Chunk3101 |
690 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations Theorem 13.2.1 If P0, P1, P2, and R are continuous and P0 and R are positive on a closed interval [a, b], then the equation P0(x)y′′ + P1(x)y′ + P2(x)y + λR(x)y = 0 (13.2.5) can be rewritten as (p(x)y′)′ + q(x)y + λr(x)y = 0, (13.2.... | Elementary Differential Equations with Boundary Value Problems_Page_700_Chunk3102 |
Section 13.2 Sturm-Liouville Problems 691 Example 13.2.4 Rewrite the eigenvalue problem x2y′′ + xy′ + λy = 0, y(1) = 0, y(2) = 0 (13.2.13) of Example 13.2.2 as a Sturm-Liouville problem. Solution Dividing the differential equation in (13.2.13) by x2 yields y′′ + 1 xy′ + λ x2 y = 0. Comparing this to (13.2.7) shows that... | Elementary Differential Equations with Boundary Value Problems_Page_701_Chunk3103 |
692 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations 1 2 3 4 1 2 3 4 k u Figure 13.2.1 u = tanh k and u = −2k/(1 −3k2) The boundary conditions require that c1 + kc2 = 0 (cosh k + 3k sinh k)c1 + (sinh k + 3k cosh k)c2 = 0. (13.2.17) The determinant of this system is DN(k) = 1 k cosh k ... | Elementary Differential Equations with Boundary Value Problems_Page_702_Chunk3104 |
Section 13.2 Sturm-Liouville Problems 693 so y′ = k(−c1 sinkx + c2 cos kx). The boundary conditions require that c1 + kc2 = 0 (cos k −3k sin k)c1 + (sin k + 3k cos k)c2 = 0. (13.2.20) The determinant of this system is DP (k) = 1 k cos k −3k sin k sin k + 3k cos k = (1 + 3k2) sin k + 2k cos k. The system (13.2.20) has a... | Elementary Differential Equations with Boundary Value Problems_Page_703_Chunk3105 |
694 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations k = π k = 2π k = 3π k = 4π k u 1 − 1 Figure 13.2.2 u = tan k and u = −2k/(1 + k) Proof Integration by parts yields Z b a [u(x)Lv(x) −v(x)Lu(x)] dx = Z b a [u(x)(p(x)v′(x))′ −v(x)(p(x)u′(x))′] dx = p(x)[u(x)v′(x) −u′(x)v(x)] b a − Z ... | Elementary Differential Equations with Boundary Value Problems_Page_704_Chunk3106 |
Section 13.2 Sturm-Liouville Problems 695 Theorem 13.2.3 If λ = p + qi with q ̸= 0 then the boundary value problem Ly + λr(x)y = 0, B1(y) = 0, B2(y) = 0 has only the trivial solution. Proof For this theorem to make sense, we must consider complex-valued solutions of Ly + (p + iq)r(x, y)y = 0. (13.2.23) If y = u + iv wh... | Elementary Differential Equations with Boundary Value Problems_Page_705_Chunk3107 |
696 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations Theorem 13.2.4 If λ1 and λ2 are distinct eigenvalues of the Sturm–Liouville problem Ly + λr(x)y = 0, B1(y) = 0, B2(y) = 0 (13.2.25) with associated eigenfunctions u and v respectively, then Z b a r(x)u(x)v(x) dx = 0. (13.2.26) Proof... | Elementary Differential Equations with Boundary Value Problems_Page_706_Chunk3108 |
Section 13.2 Sturm-Liouville Problems 697 Theorem 13.2.7 Let λ1 < λ2 < · · · < λn < · · · be the eigenvalues of the Sturm–Liouville problem Ly + λr(x)y = 0, B1(y) = 0, B2(y) = 0, with associated eigenvectors y1, y2, ..., yn, .... Suppose f is piecewise smooth (Definition 11.2.3) on [a, b]. For each n, let cn = Z b a r(x... | Elementary Differential Equations with Boundary Value Problems_Page_707_Chunk3109 |
698 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations 10. Solve the eigenvalue problem y′′ + 2y′ + y + λy = 0, y′(0) = 0, y′(1) = 0. In Exercises 11–20 : (a) Determine whether λ = 0 is an eigenvalue. If it is, find an associated eigenfunc- tion. (b) Compute the negative eigenvalues with... | Elementary Differential Equations with Boundary Value Problems_Page_708_Chunk3110 |
Section 13.2 Sturm-Liouville Problems 699 24. C 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. 25. Consider the Sturm-Liouville problem y′′ + λy = 0, y(0) = 0, y(L) + δy′(L) = 0. (A) (a) Show that (A... | Elementary Differential Equations with Boundary Value Problems_Page_709_Chunk3111 |
700 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations (b) 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. ... | Elementary Differential Equations with Boundary Value Problems_Page_710_Chunk3112 |
Section 13.2 Sturm-Liouville Problems 701 32. Show that the eigenvalues of the Sturm–Liouville problem (p(x)y′)′ + λr(x)y = 0, αy(a) + βy′(a) = 0, ρy(b) + δy′(b) are all positive if αβ ≤0, ρδ ≥0, and (αβ)2 + (ρδ)2 > 0. | Elementary Differential Equations with Boundary Value Problems_Page_711_Chunk3113 |
702 Chapter 13 Boundary Value Problems for Second Order Ordinary Differential Equations A BRIEF TABLE OF INTEGRALS Z uα du = uα+1 α + 1 + c, α ̸= −1 Z du u = ln |u| + c Z cos u du = sin u + c Z sinu du = −cos u + c Z tanu du = −ln | cos u| + c Z cot u du = ln| sin u| + c Z sec2 u du = tan u + c Z csc2 u du = −cot u + c... | Elementary Differential Equations with Boundary Value Problems_Page_712_Chunk3114 |
Section 13.2 Sturm-Liouville Problems 703 Z u sinu du = −u cos u + sin u + c Z ueu du = ueu −eu + c Z eλu cos ωu du = eλu(λ cos ωu + ω sinωu) λ2 + ω2 + c Z eλu sin ωu du = eλu(λ sin ωu −ω cos ωu) λ2 + ω2 + c Z ln|u| du = u ln|u| −u + c Z u ln|u| du = u2 ln |u| 2 −u2 4 + c Z cos ω1u cos ω2u du = sin(ω1 + ω2)u 2(ω1 + ω2)... | Elementary Differential Equations with Boundary Value Problems_Page_713_Chunk3115 |
Answers to Selected Exercises Section 1.2 Answers, pp. 14–15 1.2.1 (p. 14) (a) 3 (b) 2 (c) 1 (d) 2 1.2.3 (p. 14) (a) y = −x2 2 + c (b) y = x cos x −sin x + c (c) y = x2 2 lnx −x2 4 + c (d) y = −x cos x + 2 sin x + c1 + c2x (e) y = (2x −4)ex + c1 + c2x (f) y = x3 3 −sinx + ex + c1 + c2x (g) y = sin x + c1 + c2x + c3x2 (... | Elementary Differential Equations with Boundary Value Problems_Page_714_Chunk3116 |
Answers to Selected Exercises 705 Section 2.1 Answers, pp. 41–44 2.1.1 (p. 41) y = e−ax 2.1.2 (p. 41) y = ce−x3 2.1.3 (p. 41) y = ce−(ln x)2/2 2.1.4 (p. 41) y = c x3 2.1.5 (p. 41) y = ce1/x 2.1.6 (p. 41) y = e−(x−1) x 2.1.7 (p. 41) y = e x lnx 2.1.8 (p. 41) y = π x sin x 2.1.9 (p. 41) y = 2(1 + x2) 2.1.10 (p. 41) y = 3... | Elementary Differential Equations with Boundary Value Problems_Page_715_Chunk3117 |
706 Answers to Selected Exercises 2.1.43 (p. 43) G = r λ + G0 −r λ e−λt limt→∞G(t) = r λ 2.1.45 (p. 43) (a) y = y0e−a(x−x0) + e−ax Z x x0 eatf(t) dt 2.1.48 (p. 44) (a) y = tan−1 1 3 + ce3x (b) y = ± ln 1 x + c x2 1/2 (c) y = exp x2 + c x2 (d) y = −1 + x c + 3 ln|x| Section 2.2 Answers, pp. 52–55 2.2.1 ... | Elementary Differential Equations with Boundary Value Problems_Page_716_Chunk3118 |
Answers to Selected Exercises 707 2.2.24 (p. 53) y = x + c 1 −cx 2.2.25 (p. 53) y = −x cos c + √ 1 −x2 sin c; y ≡1; y ≡−1 2.2.26 (p. 53) y = −x + 3π/2 2.2.28 (p. 53) P = P0 αP0 + (1 −αP0)e−at ; limt→∞P (t) = 1/α 2.2.29 (p. 53) I = SI0 I0 + (S −I0)e−rSt 2.2.30 (p. 53) If q = rS then I = I0 1 + rI0t and limt→∞I(t) = 0. I... | Elementary Differential Equations with Boundary Value Problems_Page_717_Chunk3119 |
708 Answers to Selected Exercises 2.3.12 (p. 61) (a), (b) all (x0, y0) such that x0 + y0 > 0 2.3.13 (p. 61) (a), (b) all (x0, y0) with x0 ̸= 1, y0 ̸= (2k + 1)π 2 (k = integer) 2.3.16 (p. 61) y = 3 5x + 1 5/3 , −∞< x < ∞, is a solution. Also, y = ( 0, −∞< x ≤−5 3 | Elementary Differential Equations with Boundary Value Problems_Page_718_Chunk3120 |
Answers to Selected Exercises 709 is a solution of (A) on (−∞, ∞). If α ≥0, then y = 1 + (x2 −α2)3/2, −∞< x < −α, 1, −α ≤x ≤ √ 5, 1 −(x2 −5)3/2, √ 5 < x < ∞, and y = 1 −(x2 −α2)3/2, −∞< x < −α, 1, −α ≤x ≤ √ 5, 1 −(x2 −5)3/2, √ 5 < x < ∞, are also solutions of (A) on (−∞, ∞). Section 2.4 Answers, pp. 68–72 2... | Elementary Differential Equations with Boundary Value Problems_Page_719_Chunk3121 |
710 Answers to Selected Exercises 2.4.26 (p. 70) y = −x(4x −3) (2x −3) 2.4.27(p. 70) y = x √ 4x6 −1 2.4.28 (p. 70) tan−1 y x −1 2 ln(x2 + y2) = c 2.4.29 (p. 70) (x + y) ln |x| + y(1 −ln |y|) + cx = 0 2.4.30 (p. 70) (y + x)3 = 3x3(ln|x| + c) 2.4.31 (p. 70) (y + x) = c(y −x)3; y = x; y = −x 2.4.32 (p. 70) y2(y −3x) = c; ... | Elementary Differential Equations with Boundary Value Problems_Page_720_Chunk3122 |
Answers to Selected Exercises 711 2.4.57 (p. 72) y = ex − 1 1 + ce−x 2.4.58 (p. 72) y = 1 − 1 x(1 −cx) 2.4.59 (p. 72) y = x − 2x x2 + c | Elementary Differential Equations with Boundary Value Problems_Page_721_Chunk3123 |
712 Answers to Selected Exercises Section 2.5 Answers, pp. 79–82 2.5.1 (p. 79) 2x3y2 = c 2.5.2 (p. 79) 3y sinx + 2x2ex + 3y = c 2.5.3 (p. 79) Not exact 2.5.4 (p. 79) x2 −2xy2 + 4y3 = c 2.5.5 (p. 79) x + y = c 2.5.6 (p. 79) Not exact 2.5.7 (p. 79) 2y2 cos x + 3xy3 −x2 = c 2.5.8 (p. 79) Not exact 2.5.9 (p. 79) x3+x2y+4xy... | Elementary Differential Equations with Boundary Value Problems_Page_722_Chunk3124 |
Answers to Selected Exercises 713 2.5.40 (p. 82) y = −e−x2 2x + √ 9 −5x2 3 ! . 2.5.44 (p. 82) (a) G(x, y) = 2xy + c (b) G(x, y) = ex sin y + c (c) G(x, y) = 3x2y −y3 + c (d) G(x, y) = −sin x sinhy + c (e) G(x, y) = cos x sinh y + c Section 2.6 Answers, pp. 91–93 2.6.3 (p. 91) µ(x) = 1/x2; y = cx and µ(y) = 1/y2; x = cy... | Elementary Differential Equations with Boundary Value Problems_Page_723_Chunk3125 |
714 Answers to Selected Exercises 2.6.22 (p. 92) µ(x, y) = x−4y−3 (and others) xy = c 2.6.23 (p. 92) µ(x, y) = xey; x2yey sin x = c 2.6.24 (p. 92) µ(x) = 1/x2; x3y3 3 −y x = c 2.6.25 (p. 92) µ(x) = x + 1; y(x + 1)2(x + y) = c 2.6.26 (p. 92) µ(x, y) = x2y2; x3y3(3x + 2y2) = c 2.6.27 (p. 92) µ(x, y) = x−2y−2; 3x2y = cxy ... | Elementary Differential Equations with Boundary Value Problems_Page_724_Chunk3126 |
Answers to Selected Exercises 715 3.1.12 (p. 108) x h = 0.1 h = 0.05 h = 0.025 “Exact” 2.0 -0.772381768 -0.761510960 -0.756179726 -0.750912371 3.1.13 (p. 108) Euler’s method x h = 0.1 h = 0.05 h = 0.025 Exact 1.0 0.538871178 0.593002325 0.620131525 0.647231889 Euler semilinear method x h = 0.1 h = 0.05 h = 0.025 Exact ... | Elementary Differential Equations with Boundary Value Problems_Page_725_Chunk3127 |
716 Answers to Selected Exercises 3.1.17 (p. 108) Euler’s method x h = 0.0500 h = 0.0250 h = 0.0125 “Exact” 1.50 0.319892131 0.330797109 0.337020123 0.343780513 Euler semilinear method x h = 0.0500 h = 0.0250 h = 0.0125 “Exact” 1.50 0.305596953 0.323340268 0.333204519 0.343780513 3.1.18 (p. 108) Euler’s method x h = 0.... | Elementary Differential Equations with Boundary Value Problems_Page_726_Chunk3128 |
Answers to Selected Exercises 717 3.2.1 (p. 116) y1 = 1.542812500, y2 = 2.421622101, y3 = 4.208020541 3.2.2 (p. 116) y1 = 1.220207973, y2 = 1.489578775 y3 = 1.819337186 3.2.3 (p. 116) y1 = 1.890687500, y2 = 1.763784003, y3 = 1.622698378 3.2.4 (p. 116) y1 = 2.961317914 y2 = 2.920132727 y3 = 2.876213748. 3.2.5 (p. 116) y... | Elementary Differential Equations with Boundary Value Problems_Page_727_Chunk3129 |
718 Answers to Selected Exercises 3.2.15 (p. 118) Improved Euler method x h = 0.2 h = 0.1 h = 0.05 “Exact" 2.0 0.924335375 0.907866081 0.905058201 0.904276722 Improved Euler semilinear method x h = 0.2 h = 0.1 h = 0.05 “Exact" 2.0 0.969670789 0.920861858 0.908438261 0.904276722 3.2.16 (p. 118) Improved Euler method x h... | Elementary Differential Equations with Boundary Value Problems_Page_728_Chunk3130 |
Answers to Selected Exercises 719 3.2.21 (p. 118) Improved Euler method x h = 0.1 h = 0.05 h = 0.025 “Exact" 1.0 50.534556346 53.483947013 54.391544440 54.729594761 Improved Euler semilinear method x h = 0.1 h = 0.05 h = 0.025 “Exact" 1.0 54.709041434 54.724083572 54.728191366 54.729594761 3.2.22 (p. 118) Improved Eule... | Elementary Differential Equations with Boundary Value Problems_Page_729_Chunk3131 |
720 Answers to Selected Exercises 3.3.7 (p. 124) x h = 0.1 h = 0.05 h = 0.025 Exact 2.0 1.353191745 1.353193606 1.353193712 1.353193719 3.3.8 (p. 125) x h = 0.05 h = 0.025 h = 0.0125 Exact 1.50 10.498658198 10.499906266 10.499993820 10.500000000 3.3.9 (p. 125) x h = 0.1 h = 0.05 h = 0.025 h = 0.1 h = 0.05 h = 0.025 3.0... | Elementary Differential Equations with Boundary Value Problems_Page_730_Chunk3132 |
Answers to Selected Exercises 721 3.3.14 (p. 126) Runge-Kutta method x h = 0.1 h = 0.05 h = 0.025 “Exact" 3.0 15.281660036 15.281981407 15.282003300 15.282004826 Runge-Kutta semilinear method x h = 0.1 h = 0.05 h = 0.025 “Exact" 3.0 15.282005990 15.282004899 15.282004831 15.282004826 3.3.15 (p. 126) Runge-Kutta method ... | Elementary Differential Equations with Boundary Value Problems_Page_731_Chunk3133 |
722 Answers to Selected Exercises 3.3.19 (p. 126) Runge-Kutta method x h = 0.0500 h = 0.0250 h = 0.0125 “Exact" 1.50 2.244025683 2.244024088 2.244023989 2.244023982 Runge-Kutta semilinear method x h = 0.0500 h = 0.0250 h = 0.0125 “Exact" 1.50 2.244025081 2.244024051 2.244023987 2.244023982 3.3.20 (p. 126) Runge-Kutta m... | Elementary Differential Equations with Boundary Value Problems_Page_732_Chunk3134 |
Answers to Selected Exercises 723 4.1.4 (p. 138) τ ln(p0/p1) ln 2 4.1.5 (p. 138) tp tq = ln p ln q 4.1.6 (p. 138) k = 1 t2 −t1 ln Q1 Q2 4.1.7 (p. 138) 20 g 4.1.8 (p. 138) 50 ln 2 3 yrs 4.1.9 (p. 138) 25 2 ln 2% 4.1.10 (p. 138) (a) = 20 ln 3 yr (b). Q0 = 100000e−.5 4.1.11 (p. 138) (a) Q(t) = 5000 −4750e−t/10 (b) 5000 lb... | Elementary Differential Equations with Boundary Value Problems_Page_733_Chunk3135 |
724 Answers to Selected Exercises 4.2.20 (p. 150) (a) cn = c 1 −e−rt/W n−1 X j=0 1 j! rt W j! (b) c (c) 0 4.2.21 (p. 150) Let c∞= c1W1 + c2W2 W1 + W2 , α = c2W 2 2 −c1W 2 1 W1 + W2 , and β = W1 + W2 W1W2 . Then: (a) c1(t) = c∞+ α W1 e−rβt, c2(t) = c∞−α W2 e−rβt (b) limt→∞c1(t) = limt→∞c2(t) = c∞ Section 4.3 Answers,... | Elementary Differential Equations with Boundary Value Problems_Page_734_Chunk3136 |
Answers to Selected Exercises 725 30v2 + y2(12y3 −15y2 −80y + 120) = 37 4.4.9 (p. 176) No equilibria if a < 0; 0 is unstable if a = 0; √a is stable and −√a is unstable if a > 0. * 4.4.10 (p. 176) 0 is a stable equilibrium if a ≤0; −√a and √a are stable and 0 is unstable if a > 0. 4.4.11 (p. 176) 0 is unstable if a ≤0; ... | Elementary Differential Equations with Boundary Value Problems_Page_735_Chunk3137 |
726 Answers to Selected Exercises 5.1.4 (p. 204) (a) y = c1 x −1 + c2 x + 1 (b) y = 2 x −1 − 3 x + 1 ; (−1, 1) 5.1.5 (p. 204) (a) ex (b) e2x cos x (c) x2 + 2x −2 (d) −5 6x−5/6 (e) −1 x2 (f) (x ln |x|)2 (g) e2x 2√x 5.1.6 (p. 204) 0 5.1.7 (p. 204) W(x) = (1 −x2)−1 5.1.8 (p. 205) W(x) = 1 x 5.1.10 (p. 205) y2 = e−x 5.1.11... | Elementary Differential Equations with Boundary Value Problems_Page_736_Chunk3138 |
x −π 3 5.2.30 (p. 219) y = k0 r2 −r1 | Elementary Differential Equations with Boundary Value Problems_Page_736_Chunk3139 |
Answers to Selected Exercises 727 5.2.32 (p. 219) y = eλ(x−x0) h k0 cos ω(x −x0) + k1 −λk0 ω sin ω(x −x0) i Section 5.3 Answers, pp. 227–229 5.3.1 (p. 227) yp = −1 + 2x + 3x2; y = −1 + 2x + 3x2 + c1e−6x + c2ex 5.3.2 (p. 227) yp = 1 + x; y = 1 + x + e2x(c1 cos x + c2 sin x) 5.3.3 (p. 227) yp = −x + x3; y = −x + x3 + ... | Elementary Differential Equations with Boundary Value Problems_Page_737_Chunk3140 |
= e−x(1+x2) 5.4.6 (p. 235) yp = ex(−2+x+2x2) 5.4.7 (p. 235) yp = xe−x 1 6 + x 2 5.4.8 (p. 235) yp = xex(1 + 2x) 5.4.9 (p. 235) yp = xe3x −1 + x 2 5.4.10 (p. 235) yp = xe2x(−2+x) 5.4.11 (p. 235) yp = x2e−x 1 + x 2 5.4.12 (p. 235) yp = x2ex 1 2 −x 5.4.13 (p. 235) yp = x2e2x 2 (1 −x + x2) 5.4.14 (p. 235) yp ... | Elementary Differential Equations with Boundary Value Problems_Page_737_Chunk3141 |
728 Answers to Selected Exercises 5.4.15 (p. 235) y = e3x 4 (−1 + 2x) + c1ex + c2e2x 5.4.16 (p. 235) y = ex(1 −2x) + c1e2x + c2e4x 5.4.17 (p. 235) y = e2x 5 (1 −x) + e−3x(c1 + c2x) 5.4.18 (p. 235) y = xex(1 −2x) + c1ex + c2e−3x 5.4.19 (p. 235) y = ex x2(1 −2x) + c1 + c2x 5.4.20 (p. 236) y = −e2x(1 + x) + 2e−x −e5x 5... | Elementary Differential Equations with Boundary Value Problems_Page_738_Chunk3142 |
(p. 245) yp = xe3x −ex 5 (cos x −2 sin x) 5.5.28 (p. 245) yp = x(cos x + 2 sin x) −ex 2 (1 −x) + e−x 2 5.5.29 (p. 245) yp = −xex 2 (2 + x) + 2xe2x + 1 10(3 cos x + sin x) | Elementary Differential Equations with Boundary Value Problems_Page_738_Chunk3143 |
Answers to Selected Exercises 729 5.5.30 (p. 245) yp = xex(cos x + x sin x) + e−x 25 (4 + 5x) + 1 + x + x2 2 5.5.31 (p. 245) yp = x2e2x 6 (3 + x) −e2x(cos x −sin x) + 3e3x + 1 4(2 + x) 5.5.32 (p. 245) y = (1 −2x + 3x2)e2x + 4 cos x + 3 sin x 5.5.33 (p. 245) y = xe−2x cos x + 3 cos 2x 5.5.34 (p. 245) y = −3 8 cos 2x + 1... | Elementary Differential Equations with Boundary Value Problems_Page_739_Chunk3144 |
730 Answers to Selected Exercises 5.6.38 (p. 254) (a) y = −kc1 sin kx + kc2 cos kx c1 cos kx + c2 sin kx (b) y = c1 + 2c2ex c1 + c2ex (c) y = −6c1 + c2e7x c1 + c2e7x (d) y = −7c1 + c2e6x c1 + c2e6x (e) y = −(7c1 −c2) cos x + (c1 + 7c2) sin x c1 cos x + c2 sin x (f) y = −2c1 + 3c2e5x/6 6(c1 + c2e5x/6) (g) y = c1 + c2(x ... | Elementary Differential Equations with Boundary Value Problems_Page_740_Chunk3145 |
Answers to Selected Exercises 731 Section 6.1 Answers, pp. 277–278 6.1.1 (p. 277) y = 3 cos 4 √ 6t − 1 2 √ 6 sin 4 √ 6t ft 6.1.2 (p. 277) y = −1 4 cos 8 √ 5t − 1 4 √ 5 sin 8 √ 5t ft 6.1.3 (p. 277) y = 1.5 cos 14 √ 10t cm 6.1.4 (p. 277) y = 1 4 cos 8t −1 16 sin 8t ft; R = √ 17 16 ft; ω0 = 8 rad/s; T = π/4 s; φ ≈−.245 ra... | Elementary Differential Equations with Boundary Value Problems_Page_741_Chunk3146 |
732 Answers to Selected Exercises 6.2.10 (p. 287) y = e−3 2 t 1 2 cos √ 119 2 t − 9 2 √ 119 sin √ 119 2 t ft 6.2.11 (p. 287) y = e−8t 1 4 cos 8 √ 2t − 1 4 √ 2 sin 8 √ 2t ft 6.2.12 (p. 287) y = e−t −1 3 cos 3 √ 11t + 14 9 √ 11 sin 3 √ 11t ft 6.2.13 (p. 287) yp = 22 61 cos 2t + 2 61 sin 2t ft 6.2.14 (p. 288) ... | Elementary Differential Equations with Boundary Value Problems_Page_742_Chunk3147 |
Answers to Selected Exercises 733 1 −k h2 1/2 : (i) r = r0 cosh γ(θ −θ0) −r0r′ 0 γh sinh γ(θ −θ0) −1 (ii) r = r0 1 −r0r′ 0 h (θ −θ0) −1 ; (iii) r = r0 cos γ(θ −θ0) −r0r′ 0 γh sin γ(θ −θ0) −1 Section 7.1 Answers, pp. 316–319 7.1.1 (p. 316) (a) R = 2; I = (−1, 3); (b) R = 1/2; I = (3/2, 5/2) (c) R = 0; (d) R = 1... | Elementary Differential Equations with Boundary Value Problems_Page_743_Chunk3148 |
4j + 1 2j + 1 # x2m + a1 ∞ X m=0 (−1)m " m−1 Y j=0 (4j + 3) # x2m+1 2mm! | Elementary Differential Equations with Boundary Value Problems_Page_743_Chunk3149 |
734 Answers to Selected Exercises 7.2.6 (p. 328) y = a0 ∞ X m=0 (−1)m "m−1 Y j=0 (4j + 1)2 2j + 1 # x2m 8mm! + a1 ∞ X m=0 (−1)m "m−1 Y j=0 (4j + 3)2 2j + 3 # x2m+1 8mm! 7.2.7 (p. 328) y = a0 ∞ X m=0 2mm! Qm−1 j=0 (2j + 1) x2m + a1 ∞ X m=0 Qm−1 j=0 (2j + 3) 2mm! x2m+1 7.2.8 (p. 328) y = a0 1 −14x2 + 35 3 x4 + a1 x ... | Elementary Differential Equations with Boundary Value Problems_Page_744_Chunk3150 |
Answers to Selected Exercises 735 7.2.34 (p. 333) y = a0 ∞ X m=0 2 3 m "m−1 Y j=0 (3j + 2) # x3m m! + a1 ∞ X m=0 6mm! Qm−1 j=0 (3j + 4) x3m+1 7.2.35 (p. 333) y = a0 ∞ X m=0 (−1)m 3mm! Qm−1 j=0 (3j + 2) x3m + a1 ∞ X m=0 (−1)m "m−1 Y j=0 (3j + 4) # x3m+1 3mm! 7.2.36 (p. 333) y = a0(1 −4x3 + 4x6) + a1 ∞ X m=0 2m " m−1 Y... | Elementary Differential Equations with Boundary Value Problems_Page_745_Chunk3151 |
736 Answers to Selected Exercises 7.3.20 (p. 339) y = 1 −2(x −1) + 1 2(x −1)2 −1 6(x −1)3 + 5 36 (x −1)4 − 73 1080(x −1)5 + · · · 7.3.21 (p. 339) y = 2 −(x + 2) −7 2(x + 2)2 + 4 3(x + 2)3 −1 24(x + 2)4 + 1 60 (x + 2)5 + · · · 7.3.22 (p. 339) y = 2 −2(x + 3) −(x + 3)2 + (x + 3)3 −11 12(x + 3)4 + 67 60 (x + 3)5 + · · · 7... | Elementary Differential Equations with Boundary Value Problems_Page_746_Chunk3152 |
(x + 2)5 + · · · 7.3.47 (p. 341) y = 2 −x −x2 + 7 6x3 −x4 + 89 120x5 + · · · | Elementary Differential Equations with Boundary Value Problems_Page_746_Chunk3153 |
Answers to Selected Exercises 737 7.3.48 (p. 341) y = 1 + 3 2(x −1)2 + 1 6(x −1)3 −1 8 (x −1)5 + · · · 7.3.49 (p. 341) y = 1 −2(x −3) + 1 2(x −3)2 −1 6(x −3)3 + 1 4(x −3)4 −1 6 (x −3)5 + · · · Section 7.4 Answers, pp. 346–347 7.4.1 (p. 346) y = c1x−4 + c2x−2 7.4.2 (p. 346) y = c1x + c2x7 7.4.3 (p. 346) y = x(c1 + c2 ln... | Elementary Differential Equations with Boundary Value Problems_Page_747_Chunk3154 |
xn; x−1 7.5.16 (p. 358) y1 = x1/2 ∞ X n=0 (−1)n 2nn! xn; y2 = 1 x2 ∞ X n=0 (−1)n Qn j=1(2j −5)xn 7.5.17 (p. 358) y1 = x ∞ X n=0 (−1)n Qn j=1(3j + 4)xn; y2 = x−1/3 ∞ X n=0 (−1)n 3nn! xn 7.5.18 (p. 358) y1 = x ∞ X n=0 2n n! Qn j=1(2j + 1)xn; y2 = x1/2 ∞ X n=0 2n n! Qn j=1(2j −1)xn | Elementary Differential Equations with Boundary Value Problems_Page_747_Chunk3155 |
738 Answers to Selected Exercises 7.5.19 (p. 358) y1 = x1/3 ∞ X n=0 1 n! Qn j=1(3j + 2)xn; y2 = x−1/3 ∞ X n=0 1 n! Qn j=1(3j −2)xn 7.5.20 (p. 358) y1 = x 1 + 2 7x + 1 70x2 ; y2 = x−1/3 ∞ X n=0 (−1)n 3nn! n Y j=1 3j −13 3j −4 ! xn 7.5.21 (p. 358) y1 = x1/2 ∞ X n=0 (−1)n n Y j=1 2j + 1 6j + 1 ! ; xn y2 = x1/3 ∞ X n=0 ... | Elementary Differential Equations with Boundary Value Problems_Page_748_Chunk3156 |
Answers to Selected Exercises 739 7.5.40 (p. 359) y1 = x1/2 ∞ X m=0 (−1)m 4m m Y j=1 4j −1 2j + 1 ! x2m; y2 = x−1/2 ∞ X m=0 (−1)m 8mm! m Y j=1 (4j −3) ! x2m 7.5.41 (p. 359) y1 = x1/2 ∞ X m=0 (−1)m m! m Y j=1 (2j + 1) ! x2m; y2 = 1 x2 ∞ X m=0 (−2)m m Y j=1 4j −3 4j −5 ! x2m 7.5.42 (p. 359) y1 = x1/3 ∞ X m=0 (−1)m m Y j=... | Elementary Differential Equations with Boundary Value Problems_Page_749_Chunk3157 |
· 7.6.3 (p. 373) y1 = 1 + x −x2 + 1 3x3 + · · ·; y2 = y1 ln x −x 3 −1 2 x −31 18 x2 + · · · | Elementary Differential Equations with Boundary Value Problems_Page_749_Chunk3158 |
740 Answers to Selected Exercises 7.6.4 (p. 373) y1 = x1/2 1 −2x + 5 2x2 −2x3 + · · · ; y2 = y1 ln x + x3/2 1 −9 4x + 17 6 x2 + · · · 7.6.5 (p. 373) y1 = x 1 −4x + 19 2 x2 −49 3 x3 + · · · ; y2 = y1 ln x + x2 3 −43 4 x + 208 9 x2 + · · · 7.6.6 (p. 373) y1 = x−1/3 1 −x + 5 6x2 −1 2x3 + · · · ; y2 = y... | Elementary Differential Equations with Boundary Value Problems_Page_750_Chunk3159 |
Answers to Selected Exercises 741 7.6.20 (p. 374) y1 = 1 x ∞ X n=0 2n Qn j=1(2j −1) n! xn; y2 = y1 ln x + 1 x ∞ X n=1 2n Qn j=1(2j −1) n! n X j=1 1 j(2j −1) ! xn 7.6.21 (p. 374) y1 = 1 x ∞ X n=0 (−1)n Qn j=1(2j −5) n! xn; y2 = y1 ln x + 5 x ∞ X n=1 (−1)n Qn j=1(2j −5) n! n X j=1 1 j(2j −5) ! xn 7.6.22 (p. 374) y1 = x2 ... | Elementary Differential Equations with Boundary Value Problems_Page_751_Chunk3160 |
742 Answers to Selected Exercises 7.6.34 (p. 374) y1 = x−1/4 ∞ X m=0 (−1)m Qm j=1(8j −13) (32)mm! x2m; y2 = y1 ln x + 13 2 x−1/4 ∞ X m=1 (−1)m Qm j=1(8j −13) (32)mm! m X j=1 1 j(8j −13) ! x2m 7.6.35 (p. 374) y1 = x1/3 ∞ X m=0 (−1)m Qm j=1(3j −1) 9mm! x2m; y2 = y1 ln x + x1/3 2 ∞ X m=1 (−1)m Qm j=1(3j −1) 9mm! m X j=1 1... | Elementary Differential Equations with Boundary Value Problems_Page_752_Chunk3161 |
Answers to Selected Exercises 743 7.6.51 (p. 375) y1 = (1 + x2)2; y2 = y1 ln x −3 2x2 −3 2x4 + ∞ X m=3 (−1)m m(m −1)(m −2)x2m 7.6.52 (p. 375) y1 = x−1/2 1 −1 2x2 + 1 32x4 ; y2 = y1 ln x + x3/2 5 8 − 9 128 x2 + ∞ X m=2 1 4m+1(m −1)m(m + 1)(m + 1)!x2m ! . 7.6.56 (p. 377) y1 = ∞ X m=0 (−1)m 4m(m!)2 x2m; y2 = y1 ln x − ... | Elementary Differential Equations with Boundary Value Problems_Page_753_Chunk3162 |
744 Answers to Selected Exercises Section 7.7 Answers, pp. 388–390 7.7.1 (p. 388) y1 = 2x3 ∞ X n=0 (−4)n n!(n + 2)!xn; y2 = x + 4x2 −8 y1 ln x −4 ∞ X n=1 (−4)n n!(n + 2)! n X j=1 j + 1 j(j + 2) ! xn ! 7.7.2 (p. 388) y1 = x ∞ X n=0 (−1)n n!(n + 1)!xn; y2 = 1 −y1 ln x + x ∞ X n=1 (−1)n n!(n + 1)! n X j=1 2j + 1 j(j + 1) ... | Elementary Differential Equations with Boundary Value Problems_Page_754_Chunk3163 |
Answers to Selected Exercises 745 7.7.13 (p. 388) y1 = x5 ∞ X n=0 (−1)n(n + 1)(n + 2)xn; y2 = 1 −x 2 + x2 6 7.7.14 (p. 388) y1 = 1 x ∞ X n=0 (−1)n n! n Y j=1 (j + 3)(2j −3) j + 6 ! xn; y2 = x−7 1 + 26 5 x + 143 20 x2 7.7.15 (p. 388) y1 = x7/2 ∞ X n=0 (−1)n 2n(n + 4)!xn; y2 = x−1/2 1 −1 2x + 1 8x2 −1 48 x3 7.7.16 ... | Elementary Differential Equations with Boundary Value Problems_Page_755_Chunk3164 |
746 Answers to Selected Exercises 7.7.27 (p. 388) y1 = 2x3 ∞ X m=0 (−1)m 4mm!(m + 2)!x2m; y2 = x−1 1 + 1 4x2 −1 16 y1 ln x −2x3 ∞ X m=1 (−1)m 4mm!(m + 2)! m X j=1 j + 1 j(j + 2) ! x2m ! 7.7.28 (p. 388) y1 = x−1/2 ∞ X m=0 (−1)m Qm j=1(2j −1) 8mm!(m + 1)! x2m; y2 = x−5/2 + 1 4y1 ln x −x−1/2 ∞ X m=1 (−1)m Qm j=1(2j −1)... | Elementary Differential Equations with Boundary Value Problems_Page_756_Chunk3165 |
Answers to Selected Exercises 747 7.7.42 (p. 389) y1 = xν ∞ X m=0 (−1)m 4mm! Qm j=1(j + ν)x2m; y2 = x−ν ν−1 X m=0 (−1)m 4mm! Qm j=1(j −ν)x2m − 2 4νν!(ν −1)! y1 ln x −xν 2 ∞ X m=1 (−1)m 4mm!Qm j=1(j + ν) m X j=1 2j + ν j(j + ν) ! x2m ! Section 8.1 Answers, pp. 402–404 8.1.1 (p. 402) (a) 1 s2 (b) 1 (s + 1)2 (c) b s2 −b2 ... | Elementary Differential Equations with Boundary Value Problems_Page_757_Chunk3166 |
3t (b) −4 15 cos t + 1 15 sin t + 4 15 cos 4t −1 60 sin 4t | Elementary Differential Equations with Boundary Value Problems_Page_757_Chunk3167 |
748 Answers to Selected Exercises (c) 5 3 cos t + sin t −5 3 cos 2t −1 2 sin 2t (d) −1 3 cos t 2 + 2 3 sin t 2 + 1 3 cos t −1 3 sin t (e) 1 15 cos t 4 −8 15 sin t 4 −1 15 cos 4t + 1 30 sin 4t (f) 2 5 cos t 3 −3 5 sin t 3 −2 5 cos t 2 + 2 5 sin t 2 8.2.6 (p. 412) (a) et(cos 2t + sin 2t) −e−t cos 3t + 4 3 sin 3t (b) ... | Elementary Differential Equations with Boundary Value Problems_Page_758_Chunk3168 |
(p. 419) y = −(2t + 2) cos 2t + sin 2t + 3 cos t 8.3.27 (p. 419) y = et(cos t −3 sin t) + e3t 8.3.28 (p. 419) y = −1 + t + e−t(3 cos t −5 sin t) 8.3.29 (p. 419) y = 4 cos t −3 sin t −et(3 cos t −8 sin t) 8.3.30 (p. 419) y = e−t −2et + e−2t(cos 3t −11/3 sin 3t) 8.3.31 (p. 419) y = e−t(sin t −cos t) + e−2t(cos t + 4 sin ... | Elementary Differential Equations with Boundary Value Problems_Page_758_Chunk3169 |
Answers to Selected Exercises 749 8.3.36 (p. 419) y = e−t/2 10 (5t + 26) −1 5(3 cos t + sin t) 8.3.37 (p. 419) y = 1 100 | Elementary Differential Equations with Boundary Value Problems_Page_759_Chunk3170 |
750 Answers to Selected Exercises 8.4.22 (p. 428) 2 + t + u(t −1)(4 −t) + u(t −3)(t −2) = 2 + t, 0 ≤t < 1, 6, 1 ≤t < 3, t + 4, t ≥3. 8.4.23 (p. 429) 5 −t + u(t −3)(7t −15) + 3 2u(t −6)(t −6)2 = 5 −t, 0 ≤t < 3, 6t −10, 3 ≤t < 6, 44 −12t + 3 2t2, t ≥6. 8.4.24 (p. 429) u(t −π)e−2(t−π)(2 cos t −5 sin t)... | Elementary Differential Equations with Boundary Value Problems_Page_760_Chunk3171 |
Answers to Selected Exercises 751 8.5.10 (p. 437) y = t −sin t −2u(t −π)(t + sin t + π cos t) 8.5.11 (p. 437) y = u(t −2) t −1 2 + e2(t−2) 2 −2et−2 8.5.12 (p. 437) y = t + sin t + cos t −u(t −2π)(3t −3 sin t −6π cos t) 8.5.13 (p. 437) y = 1 2 + 1 2e−2t −e−t + u(t −2) | Elementary Differential Equations with Boundary Value Problems_Page_761_Chunk3172 |
752 Answers to Selected Exercises Section 8.6 Answers, pp. 448–452 8.6.1 (p. 448) (a) 1 2 Z t 0 τ sin 2(t −τ) dτ (b) Z t 0 e−2τ cos 3(t −τ) dτ (c) 1 2 Z t 0 sin 2τ cos 3(t −τ) dτ or 1 3 Z t 0 sin 3τ cos 2(t −τ) dτ (d) Z t 0 cos τ sin(t −τ) dτ (e) Z t 0 eaτdτ (f) e−t Z t 0 sin(t −τ) dτ (g) e−2t Z t 0 τeτ sin(t −τ) dτ (h... | Elementary Differential Equations with Boundary Value Problems_Page_762_Chunk3173 |
Answers to Selected Exercises 753 8.7.8 (p. 460) y = e2t + 7 cos 2t −sin 2t −1 2u(t −π/2) sin 2t 8.7.9 (p. 460) y = 1 2(1 + e−2t) + u(t −1)(e−(t−1) −e−2(t−1)) 8.7.10 (p. 460) y = 1 4et + 1 4e−t(2t −5) + 2u(t −2)(t −2)e−(t−2) 8.7.11 (p. 460) y = 1 6(2 sin t + 5 sin 2t) −1 2u(t −π/2) sin 2t 8.7.12 (p. 460) y = e−t(sin t ... | Elementary Differential Equations with Boundary Value Problems_Page_763_Chunk3174 |
754 Answers to Selected Exercises (h) x4y(4) + 3x2y′′′ −x2y′′ + 2xy′ −2y = 0 (i) (2x −1)y(4) −4xy′′′ + (5 −2x)y′′ + 4xy′ −4y = 0 (j) xy(4) −y′′′ −4xy′′ + 4y′ = 0 Section 9.2 Answers, pp. 482–487 9.2.1 (p. 482) y = ex(c1 + c2x + c3x2) 9.2.2 (p. 482) y = c1ex + c2e−x + c3 cos 3x + c4 sin 3x 9.2.3 (p. 482) y = c1ex + c2 c... | Elementary Differential Equations with Boundary Value Problems_Page_764_Chunk3175 |
e− √ 3x sin x, e √ 3x cos x, e √ 3x sin x} (f) 1, e2x, e3x/2 cos √ 3 2 x , e3x/2 sin √ 3 2 x , ex/2 cos √ 3 2 x , ex/2 sin √ 3 2 x | Elementary Differential Equations with Boundary Value Problems_Page_764_Chunk3176 |
Answers to Selected Exercises 755 (g) e−x, ex/2 cos √ 3 2 x , ex/2 sin √ 3 2 x , e−x/2 cos √ 3 2 x , e−x/2 sin √ 3 2 x 9.2.45 (p. 487) y = c1xr1 + c2xr2 + c3xr3 (r1, r2, r3 distinct); y = c1xr1 + (c2 + c3 ln x)xr2 (r1, r2 distinct); y = [c1 + c2 ln x + c3(ln x)2]xr1; y = c1xr1 + xλ[c2 cos(ω ln x) + c3 si... | Elementary Differential Equations with Boundary Value Problems_Page_765_Chunk3177 |
756 Answers to Selected Exercises 9.3.53 (p. 495) yp = 2x2(1 + x)e−x + x cos x −2 sin x 9.3.54 (p. 495) yp = 2xex + xe−x + cos x 9.3.55 (p. 495) yp = xex 6 (cos x + sin 2x) 9.3.56 (p. 495) yp = x2 54 [(2 + 2x)ex + 3e−2x] 9.3.57 (p. 495) yp = x 8 sinh x sin x 9.3.58 (p. 495) yp = x3(1 + x)e−x + xe−2x 9.3.59 (p. 495) yp ... | Elementary Differential Equations with Boundary Value Problems_Page_766_Chunk3178 |
Answers to Selected Exercises 757 9.4.32 (p. 504) y = x ln x + x −√x + 1 x + 1 √x. 9.4.33 (p. 504) y = x3 ln |x| + x −2x3 + 1 x −1 x2 9.4.35 (p. 505) yp = Z x x0 e(x−t) −3e−(x−t) + 2e−2(x−t) 6 F(t) dt 9.4.36 (p. 505) yp = Z x x0 (x −t)2(2x + t) 6xt3 F(t) dt 9.4.37 (p. 505) yp = Z x x0 xe(x−t) −x2 + x(t −1) t4 F(t) dt 9... | Elementary Differential Equations with Boundary Value Problems_Page_767_Chunk3179 |
758 Answers to Selected Exercises 10.2.1 (p. 518) (a) y′ = 2 4 4 2 y (b) y′ = −2 −2 −5 1 y (c) y′ = −4 −10 3 7 y (d) y′ = 2 1 1 2 y 10.2.2 (p. 518) (a) y′ = " −1 2 3 0 1 6 0 0 −2 # y (b) y′ = " 0 2 2 2 0 2 2 2 0 # y (c) y′ = " −1 2 2 2 −1 2 2 2 −1 # y (d) y′ = " 3 −1 −1 −2 3 2 4 −1 −2 # y 10.2.3 (p. 518... | Elementary Differential Equations with Boundary Value Problems_Page_768_Chunk3180 |
Answers to Selected Exercises 759 10.3.12 (p. 528) (b) y = 1 3 " −e−2t + e4t −10e−2t + e4t 11e−2t + e4t # (c) y = 1 3 " 2e−2t + e4t −e−2t + e4t −e−2t + e4t −e−2t + e4t 2e−2t + e4t −e−2t + e4t −e−2t + e4t −e−2t + e4t 2e−2t + e4t # k 10.3.13 (p. 528) (b) y = " 3et + 3e−t −e−2t 3et + 2e−2t −e−2t # (c) y = " e−t et −e−t 2e... | Elementary Differential Equations with Boundary Value Problems_Page_769_Chunk3181 |
760 Answers to Selected Exercises 10.4.20 (p. 540) y = " 5 5 0 # et/2 + " 0 0 1 # et/2 + " −1 2 0 # e−t/2 10.4.21 (p. 540) y = " 3 3 3 # et + " −2 −2 2 # e−t 10.4.22 (p. 540) y = " 2 −2 2 # et − " 3 0 3 # e−2t + " 1 1 0 # e3t 10.4.23 (p. 540) y = − " 1 2 1 # et + " 4 2 4 # e−t + " 1 1 0 # e2t 10.4.24 (p. 540) y = " −2 ... | Elementary Differential Equations with Boundary Value Problems_Page_770_Chunk3182 |
Answers to Selected Exercises 761 10.4.39 (p. 541) Points on L1 : y2 = y1/2 are trajectories of constant solutions. The trajectories of nonconstant solutions are half-lines on either side of L1, parallel to 1 −1 , L1. 10.4.40 (p. 541) Points on L2 : y2 = −y1 are trajectories of constant solutions. The trajectories ... | Elementary Differential Equations with Boundary Value Problems_Page_771_Chunk3183 |
762 Answers to Selected Exercises 10.5.15 (p. 554) y = 2 3 e−5t − 8 4 te−5t 10.5.16 (p. 554) y = 3 1 e5t − 12 6 te5t 10.5.17 (p. 554) y = 0 2 e−4t + 6 6 te−4t 10.5.18 (p. 554) y = " 4 8 −6 # et + " 2 −3 −1 # e−2t + " −1 1 0 # te−2t 10.5.19 (p. 554) y = " 3 3 6 # e2t − " 9 5 6 # + " 2 2 0 # t 10.... | Elementary Differential Equations with Boundary Value Problems_Page_772_Chunk3184 |
Answers to Selected Exercises 763 10.5.27 (p. 555) y = c1 " 0 1 1 # e2t + c2 " 1 1 0 # e2t 2 + " 0 1 1 # te2t ! +c3 " −1 1 0 # e2t 8 + " 1 1 0 # te2t 2 + " 0 1 1 # t2e2t 2 ! 10.5.28 (p. 555) y = c1 " −2 1 2 # e−6t + c2 − " 6 1 0 # e−6t 6 + " −2 1 2 # te−6t ! +c3 − " 12 1 0 # e−6t 36 − " 6 1 0 # te−6t 6 + " −2 1 2 # t2e... | Elementary Differential Equations with Boundary Value Problems_Page_773_Chunk3185 |
764 Answers to Selected Exercises 10.6.8 (p. 566) y = c1 " −1 1 1 # et + c2e−t " −sin 2t −cos 2t 2 cos 2t 2 cos 2t # + c3e−t " cos 2t −sin 2t 2 sin 2t 2 sin 2t # 10.6.9 (p. 566) y = c1e3t cos 6t −3 sin 6t 5 cos 6t + c2e3t sin 6t + 3 cos 6t 5 sin 6t 10.6.10 (p. 566) y = c1e2t cos t −3 sin t 2 cos t + c2e2t ... | Elementary Differential Equations with Boundary Value Problems_Page_774_Chunk3186 |
Answers to Selected Exercises 765 10.6.31 (p. 567) U ≈ .8507 .5257 , V ≈ −.5257 .8507 10.6.32 (p. 567) U ≈ −.9732 .2298 , V ≈ .2298 .9732 10.6.33 (p. 567) U ≈ .5257 .8507 , V ≈ −.8507 .5257 10.6.34 (p. 567) U ≈ −.5257 .8507 , V ≈ .8507 .5257 10.6.35 (p. 568) U ≈ −.8817 .4719 , V ≈ ... | Elementary Differential Equations with Boundary Value Problems_Page_775_Chunk3187 |
766 Answers to Selected Exercises (b) y1 y2 · · · yn y′ 1 y′ 2 · · · y′ n ... ... ... ... y(n−1) 1 y(n−1) 2 · · · y(n−1) n Section 11.1 Answers, pp. 585–586 11.1.2 (p. 585) λn = n2, yn = sin nx, n = 1, 2, 3, . . . 11.1.3 (p. 585) λ0 = 0, y0 = 1; λn = n2, yn = cos nx, n = 1, 2, 3, . . . 11.1.4 (p. 585) λn = ... | Elementary Differential Equations with Boundary Value Problems_Page_776_Chunk3188 |
Answers to Selected Exercises 767 11.2.5 (p. 598) F(x) = 2 π −4 π ∞ X n=1 1 4n2 −1 cos 2nx; F(x) = | sin x|, −π ≤x ≤π 11.2.6 (p. 599) F(x) = −1 2 sin x + 2 ∞ X n=2 (−1)n n n2 −1 sin nx;; F(x) = x cos x, −π ≤x ≤π 11.2.7 (p. 599) F(x) = −2 π + π 2 cos x −4 π ∞ X n=1 4n2 + 1 (4n2 −1)2 cos 2nx; F(x) = |x|cos x, −π ≤x ≤π 11... | Elementary Differential Equations with Boundary Value Problems_Page_777_Chunk3189 |
768 Answers to Selected Exercises F(x) = 2, x = −4, 0, −4 < x < 0, x, 0 ≤x < 4, 2, x = 4 11.2.16 (p. 599) F(x) = 1 2 + 1 π ∞ X n=1 1 n sin 2nπx + 8 π3 ∞ X n=0 1 (2n + 1)3 sin(2n + 1)πx; F(x) = 1 2 , x = −1, x2, −1 < x < 0, 1 2 , x = 0, 1 −x2, 0 < x < 1, 1 2 , x = 1 11.2.17 (p. 599) F(x) = 3 ... | Elementary Differential Equations with Boundary Value Problems_Page_778_Chunk3190 |
Answers to Selected Exercises 769 11.3.7 (p. 613) C(x) = 1 3 + 4 π2 ∞ X n=1 1 n2 cos nπx 11.3.8 (p. 613) C(x) = eπ −1 π + 2 π ∞ X n=1 [(−1)neπ −1] (n2 + 1) cos nx 11.3.9 (p. 613) C(x) = L2 6 −L2 π2 ∞ X n=1 1 n2 cos 2nπx L 11.3.10 (p. 613) C(x) = −2L2 3 + 4L2 π2 ∞ X n=1 1 n2 cos nπx L 11.3.11 (p. 614) S(x) = 4 π ∞ X n=1... | Elementary Differential Equations with Boundary Value Problems_Page_779_Chunk3191 |
770 Answers to Selected Exercises 11.3.25 (p. 614) SM(x) = 4 π ∞ X n=1 1 (2n −1) sin (2n −1)πx 2L 11.3.26 (p. 614) SM(x) = −16L2 π2 ∞ X n=1 1 (2n −1)2 (−1)n + 2 (2n −1)π sin (2n −1)πx 2L 11.3.27 (p. 614) SM(x) = 4 π ∞ X n=1 1 2n −1 1 −cos (2n −1)π) 4 sin (2n −1)πx 2L 11.3.28 (p. 614) SM(x) = 4 π ∞ X n=1 2n −1 (... | Elementary Differential Equations with Boundary Value Problems_Page_780_Chunk3192 |
Answers to Selected Exercises 771 11.3.45 (p. 615) CM(x) = −96L3 π3 ∞ X n=1 1 (2n −1)3 (−1)n + 2 (2n −1)π cos (2n −1)πx 2L 11.3.46 (p. 615) CM(x) = 96L3 π3 ∞ X n=1 1 (2n −1)3 (−1)n3 + 4 (2n −1)π cos (2n −1)πx 2L 11.3.47 (p. 615) CM(x) = 96L3 π3 ∞ X n=1 1 (2n −1)3 (−1)n5 + 8 (2n −1)π cos (2n −1)πx 2L 11.3.48... | Elementary Differential Equations with Boundary Value Problems_Page_781_Chunk3193 |
772 Answers to Selected Exercises 12.1.15 (p. 626) u(x,t) = 96 π5 ∞ X n=1 1 (2n −1)5 e−5(2n−1)2π2t sin(2n −1)πx 12.1.16 (p. 626) u(x,t) = −240 π5 ∞ X n=1 1 + (−1)n2 n5 e−2n2π2t sin nπx. 12.1.17 (p. 627) u(x,t) = 16 3 + 64 π2 ∞ X n=1 (−1)n n2 e−9π2n2t/16 cos nπx 4 12.1.18 (p. 627) u(x,t) = −8 3 + 16 π2 ∞ X n=1 1 n2 e−n2... | Elementary Differential Equations with Boundary Value Problems_Page_782_Chunk3194 |
Answers to Selected Exercises 773 12.1.33 (p. 628) u(x,t) = −64 ∞ X n=1 e−3(2n−1)2t/4 (2n −1)3 (−1)n + 3 (2n −1)π cos (2n −1)x 2 12.1.34 (p. 628) u(x,t) = −16 π ∞ X n=1 (−1)n 2n −1e−(2n−1)2t cos (2n −1)x 4 12.1.35 (p. 628) u(x,t) = −64 π ∞ X n=1 (−1)n 2n −1 1 − 8 (2n −1)2π2 e−9(2n−1)2π2t/64 cos (2n −1)πx 8 12.1... | Elementary Differential Equations with Boundary Value Problems_Page_783_Chunk3195 |
774 Answers to Selected Exercises 12.1.52 (p. 629) u(x,t) = sin πx + 8 π ∞ X n=1 (−1)n (2n + 1)(2n −3)e−(2n−1)2π2t/4 sin (2n −1)πx 2 12.1.53 (p. 629) u(x,t) = x3 −x + 3 + 32 π3 ∞ X n=1 e−(2n−1)2π2t/4 (2n −1)3 sin (2n −1)πx 2 Section 12.2 Answers, pp. 642–649 12.2.1 (p. 642) u(x,t) = 4 3π3 ∞ X n=1 (−1)n+1 (2n −1)3 sin 3... | Elementary Differential Equations with Boundary Value Problems_Page_784_Chunk3196 |
Answers to Selected Exercises 775 12.2.19 (p. 644) u(x,t) = −64 π3 ∞ X n=1 1 (2n −1)3 (−1)n + 3 (2n −1)π cos(2n −1)πt cos (2n −1)πx 2 12.2.20 (p. 644) u(x,t) = −512 3π4 ∞ X n=1 (−1)n (2n −1)4 sin 3(2n −1)πt 4 cos (2n −1)πx 4 12.2.21 (p. 644) u(x,t) = −64 π4 ∞ X n=1 1 (2n −1)4 (−1)n + 3 (2n −1)π sin(2n −1)πt cos... | Elementary Differential Equations with Boundary Value Problems_Page_785_Chunk3197 |
776 Answers to Selected Exercises 12.2.39 (p. 646) u(x,t) = 96 π3 ∞ X n=1 1 (2n −1)3 1 + (−1)n 2 (2n −1)π cos 3(2n −1)πt 2 sin (2n −1)πx 2 12.2.40 (p. 646) u(x,t) = 192 π ∞ X n=1 (−1)n (2n −1)4 cos (2n −1) √ 3 t 2 sin (2n −1)x 2 12.2.41 (p. 646) u(x,t) = 64 π4 ∞ X n=1 1 (2n −1)4 1 + (−1)n 2 (2n −1)π sin 3(2n −1... | Elementary Differential Equations with Boundary Value Problems_Page_786_Chunk3198 |
Answers to Selected Exercises 777 12.2.59 (p. 647) u(x,t) = 3t 5 −48 π5 ∞ X n=1 2 + (−1)n n5 sin nπt cos nπx 12.2.60 (p. 647) u(x,t) = 1 2 [Cf(x + at) + Cf (x −at)] + 1 2a R x+at x−at Cg(τ) dτ 12.2.63 (p. 648) (c) u(x, t) = f(x + at) + f(x −at) 2 + 1 2a Z x+at x−at g(u) du 12.2.64 (p. 649) u(x,t) = x(1 + 4at) 12.2.65 (... | Elementary Differential Equations with Boundary Value Problems_Page_787_Chunk3199 |
778 Answers to Selected Exercises 12.3.15 (p. 663) u(x,y) = −64 π ∞ X n=1 (−1)n cosh(2n −1)x/2 (2n −1)4 sinh(2n −1)/2 cos (2n −1)y 2 . 12.3.16 (p. 663) u(x,y) = −192 π4 ∞ X n=1 cosh(2n −1)πx/2 (2n −1)4 sinh(2n −1)π/2 (−1)n + 2 (2n −1)π cos (2n −1)πy 2 12.3.17 (p. 663) u(x,y) = ∞ X n=1 αn sinh nπy/a sinh nπb/a sin n... | Elementary Differential Equations with Boundary Value Problems_Page_788_Chunk3200 |
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