Solution: Given the following functions F(s), find f(t). (a)

Use the time-shifting theorem to determine [f(t)], where f(t) = [t 1 + e(t 1)] u(t 1)

If f(t) = te(ta) u(t a) et u(t a), find F(s)

Use the time-shifting theorem to determine [f(t)], where f(t) = [e(t 2) e2(t 2)] u(t 2).

Use property number 7 to find [f(t)] if f(t) = teatu(t 1)

Use property number 5 to find [f(t)] if f(t) = eatu(t 1).

Find the Laplace transform of the function f(t) = eat (t 1).

If f(t) = eat sin t, show that F(s) = ____________ (s + a) 2 + 2

If f(t) = eat, show that F(s) = ______ 1 (s + a)

Find the Laplace transform of the function f(t) = teat sin (t) (t 4)

Find F(s) if f(t) = eat sin t u(t 1)

Find F(s) if f(t) = teat u(t 4).

Use the results of property 3 and the fact that if f(t) = et sin t, then F(s) = 1(s + 1)2 + 1 to find the Laplace transform of f(t) = e2t sin 2t.

Find [f (t)] if f (t) = t 2 eat u(t 2)

4 If f(t) = t sin (t)u(t 1), find F(s)

If F(s) = (s + 1)2 (s + 3)2 (s + 2)(s + 4), find f(t)

If F(s) = (s + 2)2 (s + 1)(s + 3), find f(t).

Given the following functions F(s), find the inverse Laplace transform of each function. (a) F(s) = 2(s + 1) ____________ (s + 2)(s + 3) (b) F(s) = 10(s + 2) ____________ (s + 1)(s + 4) (c) F(s) = s 2 + 2s + 3 _____________ s(s + 1)(s + 2)

Given the following functions F(s), find f(t). (a) F(s) = ____________ s + 1 (s + 2)(s + 6) (b) F(s) = ____________ 24 (s + 2)(s + 3) (c) F(s) = ____________ 4 (s + 3)(s + 4) (d) F(s) = ____________ 10s

Given the following functions F(s), find f(t). (a) F(s) = s 2 + 7s + 12 __________________ (s + 2)(s + 4)(s + 6) (b) F(s) = (s + 3) (s + 6) ______________ s(s 2 + 10s + 24) (c) F(s) = s 2 + 5s + 12 __________________ (s + 2)(s + 4)(s + 6) (d) F(s) = (s + 3)(s + 6) ______________ s(s 2 + 8s + 12)

Given the following functions F(s), find f(t). (a) F(s) = ________ s + 3 s(s + 2)2 (b) F(s) = ________ s + 6 s(s + 2)3

Given the following functions F(s), find the inverse Laplace transform of each function. (a) F(s) = ________ s + 6 s 2 (s + 2) (b) F(s) = _____________ s + 4 (s + 1)2 (s + 3)

Given the following functions F(s), find f(t). (a) F(s) = ________ s + 8 s 2 (s + 4) (b) F(s) = _________ 1 s 2 (s + 1)2

Find f(t) if F(s) is given by the expression F(s) = (s + 1)2 __________________ s(s + 2)(s 2 + 2s + 2)

Find the inverse Laplace transform of F(s) where F(s) = es __________ s 2 + 2s + 2

Find the inverse Laplace transform of the function F(s) = es _______________ s 2 (s 2 + 1)(s 2 + 4)

Given the following functions F(s), find f(t). (a) F(s) = _____________ s + 1 s(s + 2)(s + 3) (b) F(s) = s 2 + s + 1 _____________ s(s + 1)(s + 2)

Given the following functions F(s), find f(t). (a) F(s) = s 2 + 4s + 8 _____________ (s + 1)(s + 4)2 (b) F(s) = s _____ + 4 s 2

Given the following functions F(s), find f(t). (a) F(s) = _______ s + 4 (s + 2)2 (b) F(s) = ________ s + 6 s(s + 1)2

Given the following functions F(s), find f(t). (a) F(s) = __________ 10 s 2 + 2s + 2 (b) F(s) = 10(s + 2) __________ s 2 + 4s + 5

Given the following functions F(s), find the inverse Laplace transform of each function. (a) F(s) = 10(s + 1) s 2 + 2s + 2 (b) F(s) = ___________ s + 1 s(s 2 + 4s + 5)

Given the following functions F(s), find f(t). (a) F(s) = s(s + 6) __________________ (s + 3)(s 2 + 6s + 18) (b) F(s) = (s + 4) (s + 8) _____________ s(s 2 + 8s + 32)

Given the following functions F(s), find f(t). (a) F(s) = (s + 1)(s + 3) _________________ (s + 2)(s 2 + 2s + 2) (b) F(s) = (s + 2)2 __________ s 2 + 4s + 5

Given the following functions F(s), find f(t). (a) F(s) = (s + 6) __________________ (s + 3)(s 2 + 6s + 16) (b) F(s) = (s + 4)(s + 8) ____________ s(s 2 + 4s + 8)

Given the following functions F(s), find f(t). (a) F(s) = 6s + 12 ______________________ (s 2 + 4s + 5)(s 2 + 4s + 8) (b) F(s) = s(s + 2) __________ s 2 + 2s + 2

Find the inverse Laplace transform of the following functions. (a) F(s) = es _____ s + 1 (b) F(s) = 1 e 2s s (c) F(s) = 1 es _______ s + 2

Find f (t) if F(s) is given by the following functions: (a) F(s) = 2(s + 1)es ____________ (s + 2)(s + 4) (b) F(s) = 10(s + 2)e2s ____________ (s + 1)(s + 4) (c) F(s) = ses ____________ (s + 4)(s + 8)

Find the inverse Laplace transform of the following functions. (a) F(s) = (s + 3)e s s(s + 2) (b) F(s) = e10s ____________ (s + 2)(s + 3) (c) F(s) = (s 2 + 2s + 1)e2s _______________ s(s + 1)(s + 2) (d) F(s) = (s + 1)e4s __________ s 2 (s + 2)

Find f(t) if F(s) is given by the following function: F(s) = (s + 1)es __________________ s(s + 2)(s 2 + 2s + 2)

Find the inverse Laplace transform of the function F(s) = 10s(s + 2)e4s __________________ (s + 1)2 (s 2 + 2s + 2)

Find f(t) if F(s) is given by the expression F(s) = s 2 e2s ________________________ (s 2 + 1)(s + 1)(s 2 + 2s 2)

Use Laplace transforms to solve the following differential equations. (a) dy(t) _____ dt + 3y(t) = et , y(0) = 1 (b) dy(t) ____ dt + 4y(t) = 2u(t), y(0) = 2

Solve the following integrodifferential equation using Laplace transforms. y(t) + 4 0 t y()e4(t ) d() = 2et t > 0

Solve the following integrodifferential equation using Laplace transforms. dy(t) ____ dt + 2y(t) + 0 t y()d = 1 e2t , y(0) = 0, t > 0

Solve the following differential equations using Laplace transforms. (a) dx(t) ____ dt + 4x(t) = e2t , x(0) = 1 (b) dx(t) ____ dt + 6x(t) = 4u(t), x(0) = 2

Solve the following differential equations using Laplace transforms. (a) d2 y(t) _____ dt2 + 2dy(t) _____ dt + y(t) = e2t , y(0) = y(0) = 0 (b) d2 y(t) _____ dt2 + 4y(t) ____ dt + 4y(t) = u(t), y(0) = 0, y(0) = 1

Use Laplace transforms to solve the following integrodifferential equation. dy(t) ____ dt + 2y(t) + o t y ()e2(t ) d() = 4u(t), y(0) = 1, t > 0

Find the inverse Laplace transform of the function F(s) using the convolution integral. F(s) = 1 s(s + 2)

Find f(t) using the convolution integral if F(s) = s + 2 s 2 (s + 1)

Find f(t) using convolution if F(s) is F(s) = ____________ 1 (s + 1)(s + 2)

Find f(t) using convolution if F(s) is (a) F(s) = ____________ 1 (s + 1)(s + 4) (b) F(s) = ____________ 10 (s + 1)(s + 3)2

Find the initial and final values of f(t) if F(s) is given as (a) F(s) = _______ s + 3 s(s + 2) (b) F(s) = 2(s 2 + 2s + 2) _________________ (s + 1)(s 2 + 2s + 6) (c) F(s) = s 2 + 2s + 1 __________________ (s + 4)(s + 2)(s + 6)

Determine the initial and final values of f(t) if F(s) is given by the expressions (a) F(s) = 2(s + 2) _______ s(s + 1) (b) F(s) = 2(s 2 + 2s + 6) __________________ (s + 1)(s + 2)(s + 3) (c) F(s) = 2s 2 _________________ (s + 1)(s 2 + 2s + 2

Find the final values of the time function f(t) given that (a) F(s) = 10(s + 1) ____________ (s + 2)(s 3) (b) F(s) = ______ 10 s 2 + 4

Find the final values of the time function f(t) given that (a) F(s) = 10(s + 6) _____________ (s + 2) (s + 3) (b) F(s) = __________ 2 s 2 + 4s + 8

Find the initial and final values of the time function f(t) if F(s) is given as (a) F(s) = 10(s + 2) ____________ (s + 1)(s + 4) (b) F(s) = s 2 + 2s + 2 ______________________ (s + 6)(s 3 + 4s 2 + 8s + 4) (c) F(s) = __________ 2s s 2 + 2s + 3

Find the initial and final values of f(t) if F(s) is given as (a) F(s) = _______________ s + 4 s(s + 12)(s + 3) (b) F(s) = ____________ s + 3 s(s 2 + 5s + 4) (c) F(s) = s 2 + 2s + 2 _________________ (s 2 + 4s + 5)(s + 3)

In the network in Fig. P13.57, the switch opens at t = 0. Use Laplace transforms to find o(t) for t > 0. t = 0 3 k 4 k 12 V + 100 F 2 k o(t) + C(t) + Figure P13.57

In the circuit in Fig. P13.58, the switch moves from position 1 to position 2 at t = 0. Use Laplace transforms to find (t) for t > 0. t = 0 6 k 6 k 2 1 12 V + 100 F (t) + Figure P13.58

In the network in Fig. P13.59, the switch opens at t = 0. Use Laplace transforms to find i(t) for t > 0. i(t) 6 3 3 H 12 V t = 0 + Figure P13.59

In the network in Fig. P13.60, the switch opens at t = 0. Use Laplace transforms to find iL(t) for t > 0. t = 0 3 1 A 1 H 0.5 F iL(t) Figure P13.60

The switch in the circuit in Fig. P13.61 has been closed for a long time and is opened at t = 0. Find i(t) for t > 0 using Laplace transforms. + 1 0.5 H 12 V 5 i(t) 2 F t = 0 Figure P13.61

The switch in the circuit in Fig. P13.62 opens at t = 0. Find i(t) for t > 0 using Laplace transforms. + 2 t = 0 3 2 4 2 H 12 V i(t) Figure P13.62

The switch in the circuit in Fig. P13.63 has been closed for a long time and is opened at t = 0. Find i(t) for t > 0, using Laplace transforms. + + 6 V 4 4 24 V i(t) H2 3 F 3 16 t = 0 Figure P13.63

The switch in the circuit in Fig. P13.64 has been closed for a long time and is opened at t = 0. Find i(t) for t > 0 using Laplace transforms. 4 2 4 2 H 1 F 12 V 12 V t = 0 + i(t) + Figure P13.64

The switch in the circuit in Fig. P13.65 has been closed for a long time and is opened at t = 0. Find i(t) for t > 0 using Laplace transforms. 5 5 5 1 H 20 V 0.04 F 10 V t = 0 i(t) + + Figure P13.65

In the circuit shown in Fig. P13.66, switch action occurs at t = 0. Determine the voltage o(t), t > 0 using Laplace transforms. 8 3 12 V 6 V t = 0 t = 0 o(t) + + + H1 2 F 1 4 Figure P13.66