# Assignment_4_solutions.pdf - ECE 205 Winter 2017 Assignment...

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ECE 205 - Winter 2017Assignment 4 SolutionsDue 12-02-20171: Definition of Laplace transformUse the definition to evaluate the Laplace transform of the following functions:a)y(t) =t.b)y(t) =eαt.c)y(t) =eαtcos(ωt).(a)L[y(t)] =L[t] =R0te-stdt= limz→∞zR0te-stdt.zR0te-stdt=1-ste-st|z0+1szR0e-stdt=1s2-ze-szs-e-sz.Finally,L[t] = limz→∞1s2-ze-szs-e-sz=1s2,provided thats >0.(b)L[eαt] =R0eαte-stdt= limz→∞zR0e(α-s)tdt.zR0e(α-s)tdt=1α-se(α-s)t|z0=1α-se(α-s)z-1.Finally,L[eαt] = limz→∞1α-se(α-s)z-1=1s-α, provided thats > α.(c)L[eαtcos(ωt)] = limz→∞zR0e(α-s)tcos(ωt)dt.LetI(z) =zR0e(α-s)tcos(ωt). We perform integration by parts twice and solve forI(z). We obtain:I(z) =e(α-s)zcos (wz)α-e(α-s)zcos (wz)s+we(α-s)zsin (wz)-α+sα2-2α s+s2+w2Finally,L[eαtcos(ωt)] = limz→∞I(z) =(s-α)(s-α)2+w2, provideds > α.2: Laplace transformsDetermine the Laplace transform of the following functions:a)y(t) = 5 sin(2t) + 2 cos(5t).b)y(t) = cos(ωt+φ).c)y(t) =e4tt5.d)y(t) =H(t-2)t3.ss2+25, by linearity of the Laplace transform.
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ECE 205 - Winter 2017Assignment 4 Solutions
3: Inverse Laplace transformFind the inverse Laplace transform of the following functions:a)Y(s) =1(s+a)(s+b).b)Y(s) =1(s2+a2)(s2+b2).c)Y(s) =1s2-4s+ 8.d)Y(s) =e-4ss2+ 4.(a)Y(s) =1(s+a)(s+b)=1b-ah11s+bi, ifb-a6
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