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mathematics
a first course in differential equations

A First Course in Differential Equations with Modeling Applications 10th edition Dennis G. Zill - Solutions

Find either F(s) or f (t), as indicated.ℒ{e2t(t – 1)2}
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.ℒ{1* t3}Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of exponential order, then L{f* g} = £{f(1)) L{g(1)} = F(s) G(s). %3D
Find either F(s) or f (t), as indicated. 2s – 1] L- [s(s + 1)J
Fill in the blanks or answer true or false. L-
Solve (5) when E = 60 V, L = 2 h, R = 50 Ω, C = 10-4 f, i1(0) = 0, and i2(0) = 0.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 L- s2 + 2s – 3
Use Definition 7.1.1 to find ℒ{f (t)}.f(t) = t sin tDefinition 7.1.1Let f be a function defined for t = 0. Then the integral
In some instances the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. In Problems 17 and 18 use Theorem 7.4.1 to reduce the given differential equation to a linear first-order DE in the transformed function Y(s) = ℒ{y(t)}. Solve the
Find either F(s) or f (t), as indicated. 5s |(s – 2)
Fill in the blanks or answer true or false. -5s ||
Solve (5) when E = 60 V, L = ½ h, R = 50 Ω, C = 10-4 f, i1(0) = 0, and i2(0) = 0.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 s+ 1 -1. g?- 4s)
Use Definition 7.1.1 to find ℒ{f (t)}.f(t) = t cos tDefinition 7.1.1Let f be a function defined for t = 0. Then the integral
In some instances the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. In Problems 17 and 18 use Theorem 7.4.1 to reduce the given differential equation to a linear first-order DE in the transformed function Y(s) = ℒ{y(t)}. Solve the
Find either F(s) or f (t), as indicated. L- (s + 1)2
Fill in the blanks or answer true or false. - 10s + 29]
Solve the system given in (17) of Section 3.3 when R1 = 6 Ω, R2 = 5Ω, L1 = 1 h, L2 = 1 h, E(t) = 50 sin t V, i2(0) = 0, and i3(0) = 0.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 1 L- s? + 3s
Use Definition 7.1.1 to find ℒ{f (t)}.f(t) = et cos tDefinition 7.1.1Let f be a function defined for t = 0. Then the integral
In problem use a graphing utility to graph the indicated solution. y(t) of Problem 14 for 0 ≤ t Problem 14 In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. y'' + y = f(t), y(0) = 1, y'(0) = 0,
Find either F(s) or f (t), as indicated. 2s + 5 L s? + 6s + 34
Fill in the blanks or answer true or false. L- [s² - 5] -5,
(a) You were asked to show that the currents i2(t) and i3(t) in the electrical network shown in Figure 7.6.8 satisfySolve the system if R1 = 10Ω, R2 = 5 Ω L = 1 h, C = 0.2 f,i2(0) = 0, and i3(0) = 0.(b) Determine the current i1(t). diz diz L + L + Riz = E(t) dt dt -R, diz dt 1 + R; + i diz C3 =
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 s + 1 L- |s² + 2 -1,
Use Definition 7.1.1 to find ℒ{f (t)}.f(t) = e-t sin tDefinition 7.1.1Let f be a function defined for t = 0. Then the integral
In problem use a graphing utility to graph the indicated solution. y(t) of Problem 13 for 0 ≤ t Problem 13 In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. y" + 16y = f(t), y(0) = 0, y'(0) = 1,
Find either F(s) or f (t), as indicated. s2 + 4s + 5
Fill in the blanks or answer true or false. 1 (s - 5)3
(a) Show that the system of differential equations for the currents i2(t) and i3(t) in the electrical network shown in Figure 7.6.7 is(b) Solve the system in part (a) if R = 5Ω, L1 = 0.01 h, L2 = 0.0125 h, E = 100 V, i2(0) = 0, and i3(0) = 0.(c) Determine the current i1(t). di, + Riz + Riz = E(t)
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 [2s -1. s2 + 9
Someone tells you that the solutions of the two IVPs
Use Definition 7.1.1 to find ℒ{f (t)}.f(t) = t2e-2tDefinition 7.1.1Let f be a function defined for t = 0. Then the integral
In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. y'' + y = f(t), y(0) = 1, y'(0) = 0, where Appendix III (1, f(t) = [sin t, 0st< /2 t2 /2
Find either F(s) or f (t), as indicated. 1 L 2 + 2s + 5
Fill in the blanks or answer true or false. 3s -
Derive the system of differential equations describing the straight-line vertical motion of the coupled springs shown in Figure 7.6.6. Use the Laplace transform to solve the system when k1 = 1, k2 = 1, k3 = 1, m1 = 1, m2 = 1 and x1(0) = 0, x1'(0) = -1, x2(0) = 0, x'2(0) = 1. ---- X2 = 0-
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 L 4s2 + 1
Solve the differential equation in Problem 13 subject to y(0) = 0, y'(0) = 0, y(L) = 0, y'(L) = 0. In this case the beam is embedded at both ends. See Figure 7.5.5. wo -L- 'y
Use Definition 7.1.1 to find ℒ{f (t)}.f(t) = te4tDefinition 7.1.1Let f be a function defined for t = 0. Then the integral
In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed.y" + 16y = f(t), y(0) = 0, y'(0) = 1, whereAppendix III f(t) = cos 4t, 0t < 0,
Find either F(s) or f (t), as indicated. s2 – 6s + 10
Fill in the blanks or answer true or false. L- [3s- 1
Solve system (1) when k1 = 3, k2 = 2, m1 = 1, m2 = 1 and x1(0) = 0, x'1(0) = 1, x2(0) = 1, x'2(0) = 0.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 4s [4s² + 1] .2
A uniform beam of length L carries a concentrated load w0 at x = ½ L. The beam is embedded at its left end and is free at its right end. Use the Laplace transform to determine the deflection y(x) fromwhere y(0) = 0, y'(0) = 0, y''(L) = 0, and y'''(L) = 0. EI wo8(x - L). dx
Use Definition 7.1.1 to find ℒ{f (t)}.f(t) = e-2t-5Definition 7.1.1Let f be a function defined for t = 0. Then the integral
In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed.y'' + y = sin t, y(0) = 1, y'(0) = -1Appendix III-1 7. sin kt 8. cos kt s + k 2k2 9. sin? kt s(s + 4k?) 2 + 2k2 10. cos? kt s(s? + 4k2) 11." class="fr-fic
Find either F(s) or f (t), as indicated. 1 ((s – 1)*J -1.
Fill in the blanks or answer true or false. L{sin 2t U(t – )} =
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 10s [s² + 16]
Use the Laplace transform to solve the given system of differential equations. dx = 4x – 2y + 2U(t – 1) dt dy = 3x - y + U(t dt U(t – x(0) = 0, y(0) =
Use the Laplace transform to solve the given initial-value problem.y" - 7y' + 6y = et + δ(t - 2) + δ(t - 4), y(0) = 0, y'(0) = 0
Use Definition 7.1.1 to find ℒ{f (t)}.f(t) = et+7Definition 7.1.1Let f be a function defined for t = 0. Then the integral
In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. y" + 9y = cos 3t, y(0) = 2, y'(0) = 5 Appendix III = F(s) 1. 1 2. t 3. 1" n! na positive integer 4. 1/2 5. 12 2532 I(a + 1) 6. 1a a >-1 7. sin kt 8. cos
Find either F(s) or f (t), as indicated. 1 -1. (s+2)
Fill in the blanks or answer true or false.ℒ{t sin 2t} = _____
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 5 L- + 49
Use the Laplace transform to solve the given system of differential equations. d'x dy dr + 3- + 3y = 0 dt d?x + 3y = te dr x(0) = 0, x'(0) = 2, y(0) = 0
In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed.y' - y = tet sin t, y(0) = 0Appendix III
Use the Laplace transform to solve the given initial-value problem.y" + 4y' + 13y = δ(t – π) + δ(t – 3π), y(0) = 1, y'(0) = 0
Find either F(s) or f (t), as indicated. O - 41 + 10 sin
Fill in the blanks or answer true or false.ℒ{e-3t sin 2t} = _____
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 1 L [5s – 2)
Use the Laplace transform to solve the given system of differential equations.
Use the Laplace transform to solve the given initial-value problem.y" + 2y' + y = δ(t - 1), y(0) = 0, y'(0) = 0
Use Definition 7.1.1 to find ℒ{f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integralis said to be the Laplace transform of f, provided that the integral converges. f()
In problem use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms in Appendix III as needed. y' + y = t sin t, y(0) = 0 Appendix III = F(s) 1. 1 2. t 3. 1" n! na positive integer 4. 1/2 5. 12 2532 I(a + 1) 6. 1a a >-1 7. sin kt 8. cos kt s + k
Find either F(s) or f (t), as indicated.ℒ{(1 – et + 3-4t)cos 5t}
Fill in the blanks or answer true or false.ℒ{sin 2t} = _____
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 1 (4s + 1
Use the Laplace transform to solve the given system of differential equations. d'x, d²y dr dr d'x d?y 4t df dr x(0) = 8, x'(0) = 0, y(0) = 0, y'(0) = 0
Use the Laplace transform to solve the given initial-value problem.y'' + 4y' + 5y = δ(t – 2π), y(0) = 0, y'(0) = 0
Use Definition 7.1.1 to find ℒ{f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integralis said to be the Laplace transform of f, provided that the integral converges. (2, 2) 1+
In problem use Theorem 7.4.1 to evaluate the given Laplace transform.ℒ{te-3 cos 3t}Theorem 7.4.1If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then
Find either F(s) or f (t), as indicated.ℒ{e-2t cos 4t}
Fill in the blanks or answer true or false.ℒ{te-7t} = _____
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 1 s + 8
Use the Laplace transform to solve the given system of differential equations. d'x d?x, dx dt dy = 0 df dt d'y, dy dx 4 dt dt dr 0 = x(0) = 1, x'(0) = 0, y(0) = -1, y'(0) = 5 + +
Use the Laplace transform to solve the given initial-value problem.y" - 2y' = 1 + δ(t - 2), y(0) = 0, y'(0) = 1
Use Definition 7.1.1 to find ℒ{f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integral
In problem use Theorem 7.4.1 to evaluate the given Laplace transform.ℒ{te2t sin 6t}Theorem 7.4.1If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then
Find either F(s) or f (t), as indicated.ℒ{et sin 3t}
Fill in the blanks or answer true or false.ℒ{e-7t} = _____
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 S S- 2 +
Use the Laplace transform to solve the given system of differential equations. d²x + x - y = 0 dr d'y + y - x = 0 x(0) = 0, x'(0) = -2, y(0) = 0, y'(0) = 1
Use the Laplace transform to solve the given initial-value problem.y'' + 2y' = δ(t - 1), y(0) = 0, y'(0) = 1
Use Definition 7.1.1 to find ℒ{f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integralis said to be the Laplace transform of f, provided that the integral converges. fo. 0st< /2 f(t) = cos t, t2 /2
In problem use Theorem 7.4.1 to evaluate the given Laplace transform.ℒ{t2 cos t}Theorem 7.4.1If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then
If ℒ{f(t)} = F(s) and ℒ{g(g)} = G(s), then ℒ-1{F(s)G(s)} = f(t)g(t). _____Fill in the blanks or answer true or false.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 L (s +2)2] + 2)
Use the Laplace transform to solve the given system of differential equations. dx + x dt dy + y = 0 dt dx dy + 2y = 0 dt dt x(0) = 0, y(0) = 1
Use the Laplace transform to solve the given initial-value problem.y'' + y = δ(t – 2π) + δ(t – 4π), y(0) = 1, y'(0) = 0
F(s) = s2/(s2 + 4) is not the Laplace transform of a function that is piecewise continuous and of exponential order. _______Fill in the blanks or answer true or false.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 + 1)3 is
Use the definition of the Laplace transform to find ℒ{f(t)}.Fill in the blanks or answer true or false.
Use a CAS to graph P1(x), P2(x), . . . , P7 (x) on the interval [1, 1].
In problem use an appropriate infinite series method about x 0 to find two solutions of the given differential equation.2xy" + y' + y = 0
(a) Use the explicit solutions y1(x) and y2(x) of Legendre’s equation given in (29) and the appropriate choice of c0 and c1 to find the Legendre polynomials P6 (x) and P7 (x).(b) Write the differential equations for which P6 (x) and P7 (x) are particular solutions.
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.x(x2 + 1)2y" + y = 0
Use Definition 7.1.1 to find ℒ{f (t)}.Definition 7.1.1Let f be a function defined for t = 0. Then the integralis said to be the Laplace transform of f, provided that the integral converges. [o, lcos t, 0st< /2 0 st< f(t) = t2 /2

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