Questions and Answers of A First Course in Differential Equations

In problem use the Laplace transform to solve the given initial value problem.2y'' + 3y'' - 3y' - 2y = e-t, y(0) = 0, y'(0) = 0, y''(0) = 1
Find ℒ{f (t)} by first using a trigonometric identity.f(t) = sin(4t + 5)
In Problem use the Laplace transform to solve the given integral equation or integrodifferential equation. f(t) = 21 – 4 si sin Tf(t – 7) dr
Find either F(s) or f (t), as indicated. L{e?-1 U(t – 2)}
Use the Laplace transform to solve the given equation.y'' + 5y + 4y = f(t), y(0) = 0, y'(0) = 3 f(t) = 12 (-1)* U(t – k). よ=0
In problem use the Laplace transform to solve the given initial value problem.y'' + 9y = et, y(0) = 0, y'(0) = 0
In Problem use the Laplace transform to solve the given integral equation or integrodifferential equation. f(t) + T) f(7) dr = t
Find either F(s) or f (t), as indicated. L{(t – 1)U(t – 1)}
Use the Laplace transform to solve the given equation.y' + 2y = f(t), y(0) = 1, where f(t) is given in Figure 7.R.10. f() 4 3.
In problem use the Laplace transform to solve the given initial value problem.y'' + y = √2 sin √2t, y(0) = 10, y'(0) = 0
Find ℒ{f (t)} by first using a trigonometric identity.f(t) = cos2t
Find {f (t)} by first using a trigonometric identity.f(t) = sin 2t cos 2t
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = e-t cosh tTheorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
Use the Laplace transform and the results of Problem 35 to solve the initial-value problemy'' + y = sin t + t sin t, y(0) = 0, y'(0) = 0Use a graphing utility to graph the solution.
Use the Laplace transform to find the charge q(t) in an RC series circuit when q(0) = 0 and E(t) = E0e-kt, k > 0. Consider two cases: k ≠ 1/RC and k = 1/RC.
Use the Laplace transform to solve the given equation.y' - 5y = f(t), where 2. 0st
In problem use the Laplace transform to solve the given initial value problem.y'' - 4y' = 6e3t - 3e-t, y(0) = 1, y'(0) = -1
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = et sinh tTheorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
The table in Appendix III does not contain an entry for(a) Use (4) along with the results in (5) to evaluate this inverse transform. Use a CAS as an aid in evaluating the convolution integral.(b)
Consider a battery of constant voltage E0 that charges the capacitor shown in Figure 7.3.9. Divide equation (20) by L and define 2λ = R/L and ω2 = 1/LC.Use the Laplace transform to show that the
USE LAPLACE TRANSFORMS TO SOLVE THE GIVEN EQUATION. y" + 6y' + 5y = t – tU (t – 2), y(0) = 1, y/ (0) = 0
In problem use the Laplace transform to solve the given initial value problem.y'' + 5y' + 4y = 0, y(0) = 1, y'(0) = 0
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = cosh ktTheorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use (8) to evaluate the given inverse transform. -1, [s(s – a)
Recall that the differential equation for the instantaneous charge q(t) on the capacitor in an LRC-series circuit is given bySee Section 5.1. Use the Laplace transform to find q(t) when L = 1 h, R =
Use the Laplace transform to solve the given equation.y'' - 8y' + 20y = tet, y(0) = 0, y'(0) = 0
In problem use the Laplace transform to solve the given initial value problem.y' – y = 2 cos 5t, y(0) = 0
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = sinh ktTheorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use (8) to evaluate the given inverse transform. 1 L- [s(s – 1))
A 4-pound weight stretches a spring 2 feet. The weight is released from rest 18 inches above the equilibrium position, and the resulting motion takes place in a medium offering a damping force
Use the Laplace transform to solve the given equation.y'' - 2y' + y = et, y(0) = 0, y'(0) = 5
In problem use the Laplace transform to solve the given initial value problem.y' + 6y = e4t, y(0) = 2
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = cos 5t + sin 2tTheorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k
In problem use (8) to evaluate the given inverse transform. 1 -1 s(s – 1)J S(S
In problem use the Laplace transform and the procedure outlined in Example 10 to solve the given boundary-value problem.y'' + 8y' + 20y = 0, y(0) = 0, y'(π) = 0
Express f in terms of unit step functions. Find ℒ{f(t)} and ℒ{etf(t)}. 1.
In problem use the Laplace transform to solve the given initial value problem.2dy/dt + y = 0, y(0) = -3
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = 4t2 - 5 sin 3tTheorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k
In problem use (8) to evaluate the given inverse transform. L- [s(s – 1)] 1,
In problem use the Laplace transform and the procedure outlined in Example 10 to solve the given boundary-value problem.y'' + 2y' + y = 0, y'(0) = 2, y(1) = 2
Express f in terms of unit step functions. Find ℒ{f(t)} and ℒ{etf(t)}. (3, 3) 2- 1+ + 1 2 3
In problem use the Laplace transform to solve the given initial value problem.dy/dt - y = 1, y (0) = 0
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = (et – e-t)2Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of
In problem use the Laplace transform to solve the given initial-value problem.y" - 2y' + 5y = 1 + t, y(0) = 0, y'(0) = 4
Express f in terms of unit step functions. Find ℒ{f(t)} and ℒ{etf(t)}. y= sin t, n
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 6s + 3 L-1 s + 5s? + 4]
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = (1 + e2t)2Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of
In problem use the Laplace transform to solve the given initial-value problem.y" - y' = e cos t, y(0) = 0, y'(0) = 0
Express f in terms of unit step functions. Find ℒ{f(t)} and ℒ{etf(t)}. 1+ + 4) 3. 2.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 (a) 1 = L-1 n! (b) " = L- n = 1, 2, 3, ... (c) eat = L-1 %3D k (d) sin kt = L-1 (e) cos
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = t2 – e-9t + 5Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of
In problem use the Laplace transform to solve the given initial-value problem.2y" + 20y' + 51y = 0, y(0) = 2,y'(0) = 0
Use the unit step function to find an equation for each graph in terms of the function y = f (t), whose graph is given in Figure 7.R.1. y=f(t) to
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 1 L
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = 1 + e4tTheorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of
In problem use the Laplace transform to solve the given initial-value problem.y" - 6y' + 13y = 0, y(0) = 0, y'(0) = -3
Use the unit step function to find an equation for each graph in terms of the function y = f (t), whose graph is given in Figure 7.R.1. y=f(t) to
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 2s – 4 ((s? + s)(s? + 1).
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = (2t - 1)3Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of
In problem use the Laplace transform to solve the given initial-value problem.y" - 4y' + 4y = t3, y(0) = 1, y'(0) = 0
Use the unit step function to find an equation for each graph in terms of the function y = f (t), whose graph is given in Figure 7.R.1. y=f(t) to
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 (s + 2)(s + 4)
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = (t + 1)3Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of
In problem use the Laplace transform to solve the given initial-value problem.y" - 6y' + 9y = t, y(0) = 0, y'(0) = 1
Use the unit step function to find an equation for each graph in terms of the function y = f (t), whose graph is given in Figure 7.R.1. y=f(t) to
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 (a) 1 = L-1 n! (b) " = L- n = 1, 2, 3, ... (c) eat = L-1 %3D k (d) sin kt = L-1 (e) cos
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = -4t2 + 16t + 9Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of
In problem use the Laplace transform to solve the given initial-value problem.y" - 4y' + 4y = t3e2t, y(0) = 0, y'(0) = 0
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 2 + 1 L s(s – 1)(s + 1)(s – 2)
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = t2 + 6t - 3Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞) and of
In problem use the Laplace transform to solve the given initial-value problem.y" + 2y' + y = 0, y(0) = 1, y'(0) = 1
Fill in the blanks or answer true or false.If ℒ{f(t) = F(s), then k > 0} then ℒ{eatf(t – k)????(t – k)} = _________.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 (s – 2)(s – 3)(s - 6)
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = 7t + 3Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.ℒ{e2t*sin t}Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞)
In problem use the Laplace transform to solve the given initial-value problem.y' - y = 1 + tet y(0) = 0
Fill in the blanks or answer true or false.If ℒ{f(t) = F(s), then ℒ{te8t f(t)} = _______.
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 s - 3 V3)(s + V3)
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = 4t - 10Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f)
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.ℒ{e-t * et cos t}Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0,
In problem use the Laplace transform to solve the given initial-value problem.y' + 4y = e4t, y(0) = 2
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 0.9s (s - 0.1)(s + 0.2)
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = t5Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f) L{sinh
In problem use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the integral before transforming.ℒ{t2 * tet}Theorem 7.4.2If f(t) and g(t) are piecewise continuous on [0, ∞)
Find either F(s) or f (t), as indicated. |(s + 1)2) (s + 2)4
Fill in the blanks or answer true or false. 1 [Ls? + n?
(a) Use the Laplace transform and the information given in Example 3 to obtain the solution (8) of the system given in (7).(b) Use a graphing utility to graph θ1(t) and θ2(t) in the tθ-plane.
(a) Show that the system of differential equations for the charge on the capacitor q(t) and the current i3(t) in the electrical network shown in Figure 7.6.9 is(b) Find the charge on the capacitor
In problem use appropriate algebra and Theorem 7.2.1 to find the given inverse Laplace transform.Theorem 7.2.1 1 [s² + s - 20
In problem use Theorem 7.1.1 to find ℒ [1]{f(t)}.f(t) = 2t4Theorem 7.1.1 1 (a) L{1} = - n! n = 1, 2, 3, ... =-- s - a (b) L{t"} (c) L{e"} k (d) L{sin kt} (e) L{cos kt} 5? + k? s? + k? k (f) L{sinh
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. a b

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