Mumbai university > Electronics and telecommunication engineering, Electronics engineering > Sem 3 > Applied mathematics 3

Marks : 06

Years : DEC 2015

(i) $L^{-1} [\log \dfrac {s^2+1}{s(s+1)}] =-\dfrac 1t L^{-1} [\dfrac d{ds} [\log \dfrac {s^2+1}{s(s+1)}]]$

$=-\dfrac 1t L^{-1} [\dfrac d{ds}[\log (s^2+1)-\log s-\log(s+1)]] \\ =-\dfrac 1t L^{-1}[\dfrac {2s}{s^2+1 } -\dfrac 1s -\dfrac 1{s+1} ] =-\dfrac 1t[2\cos t-1-e^{-t}] = (\dfrac {1+e^{-t}-2\cos t}t) $

(ii) $L^{-1}[\dfrac {(s+2)}{(s^2-2s+17)}] = L^{-1} [\dfrac {(s+2)}{(s-1^2)+16}]_{s\to s +3 } = e^tL^{-1} [\dfrac {s+3}{s^2+16}] \\ =e^t[\cos 4t + \dfrac 34\sin (4t)]$