Find the Laplace transform, F(s), for each f(t) given below for t 0-. Express F(

Question

Find the Laplace transform, F(s), for each f(t) given below for t 0-. Express F(s) as a single ratio of polynomials in s where the denominator polynomial is monic I.e., it has the value "1" (one) as the leading coefficient on the highest power of s. Your answer to each part must be in the following form: bmsn + bm-1sm-1 + ... + b2s + b0/sn + an-1sn-1 + ... + a1s + a0 Your final answers must be devoid of imaginary numbers or else lose 50 %. f(t) = 5e-1/0.04 - 6e-1/0.02 + 7 f(t) = 170sin(2pi60t-pi/5) f(t) = 9 - 3e-t cos(5t) f(t) = t2 e-4t + delta(t)

Explanation / Answer

LT(e^t) = 1/(s-1)
LT(k) = k/s
LT (sinwt) = w/(s^2 +w^2)
LT(coswt) = s/(s^2 +w^2)

a) LT(f(t)) = 5/(s+1/0.04) -6 /(s+1/0.02) +7/s
=5/(s+25) -6/(s+50) +7/s

= (6s^2 +625s +8750)/(s^3 + 75s^2 +1250s)

b) f(t) = 170[sin (120t) * cos/5 - cos(120t) * sin/5]

= 170 * 0.81 * sin 120t - 170 * 0.59 * cos 120t

=137.7 sin 120t - 99.9 cos 120t

now LT(f(t)) = 137.7 * 120/(s2 + (120)2) - 99.9 s /(s2 + (120)2)

= 51911.67/(s2 + 144000) -99.9 s/((s2 + 144000) = (-99.9s + 51911.67)/(s2 + 144000)

c)LT(e^at * cos wt) = (s-a)/((s-a)^2 +w^2)

so LT(f(t)) = 9/s - 3 (s+8)/((s+8)^2 +5^2) = 9/s -(3s+24)/(s^2 +16s+89)

= (9s^2+144s +801 - 3s^2 -24s)/(s^3 + 16s^2+89s)

= (6s^2+120s +801)/(s^3 + 16s^2+89s)

d) LT(t^2) = 2/s^3

so LT( e^(-4t) * t^2) = 2/(s+4)^3

LT(delta(t)) = 1

=> LT(f(t)) =  2/(s+4)^3 +1

=> (s^3 + 12s^2 + 48s +66)/(s^3 + 12s^2 + 48s+ 64)

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