tion 4 35+2 52 +2s +10 by the First Translation Theorem. Evaluate 코 L{eatf(t)} = F(s...
5s? +8s +2 (10 points: 5+5) Consider a function: F(s) = 2. 2s° + 2s +s (a) Use the inverse Laplace transformation technique and obtain f(t). (b) Use the final value theorem and obtain the final value lim f (t). Evaluate the result of (a) in the time domain and confirm that both answers agree.
Evaluate L{(3t+1)U(t – 2)}. 1 -2s Evaluate L s2 (s–1)
L{t"f(t)} = (-114 d Fis). Use Theorem 7.4.1. THEOREM 7.4.1 Derivatives of Transforms If F(s) = L{f(t)} and n = 1, 2, 3, .., then ds" Evaluate the given Laplace transform. (Write your answer as a function of s.) L{t cos(50)}
4. Consider the transfer function, Y($) F(S) 3 S(52 +2s + 4) (a) Qualitatively, what is the time response y(t) if f(t) represents a unit-step input? What is the value of y(t) when time is sufficiently large? What is the time constant that we may use to evaluate the "speed" of response? (b) Repeat step (a) if f(t) represents an impulse input. What is y(t) when time is sufficiently large?
Use Theorem 7.4.1. THEOREM 7.4.1 Derivatives of Transforms If F(s) = L{f(t)} and n = 1, 2, 3, ..., then L{"f(t)} = (-1)". F(s). Evaluate the given Laplace transform. (Write your answer as a function of s.) L{t cos(7t)} Find the general solution of the given differential equation. +3y= -* YOU) - Give the largest interval over which the general solution is defined. (Think about the implications of any singular points. Enter your answer using interval notation.) Determine whether there...
f(t) F(S) (s > 0) S (s > 0) n! t" ( no) (s > 0) 5+1 T(a + 1) 1a (a > -1) (s > 0) $4+1 (s > 0) S-a 1. Let f(t) be a function on [0,-). Find the Laplace transform using the definition of the following functions: a. X(t) = 7t2 b. flt) 13t+18 2. Use the table to thexight to find the Laplace transform of the following function. a. f(t)=t-4e2t b. f(t) = (5 +t)2...
Determine Laplace Transform of f(t) = u(t – 2)u(t – 3). [hint: L[u(t)] => e3s 2s e38 e-35 s e-35 2s
= 0 and L{f} = (s2 + 2s +5)(s - 1) A function f(t) has the following properties: f Ps – 10 is an unknown constant. Determine the value of P and find the function f(t). 28 +5)(-1): Where P
2 + 3s +2 (2s + 9)e-38 20. If F(s) = ? (S-2)(2+4)52+45 + 13 then L-'[F(s) = 2e2+ i sin 2x, OSI<3 (a) f(x) = { 2e2(-3) cos 3.- 3) + 2(1-3) sin 3(2x - 3), 1 3 2e22 + 3 cos 21, 0<x<3 2e2+ + 3 cos 2x + 2e-22 cos 3.0 + e -2- sin 3r, r>3 2e2+ + 2 sin 21, 05x<3 2e2+ sin 2r +2e-2(-3) cos 3(x - 3) + e -2(2-3) sin 3(-3), 1...
Find F(s). L{(1 - e* + 3e-4t) cos(2t)} S 5-1 F(s) II + 52 + 52 - 2s + 5 3s +12 52 +85 + 20 x