$\dfrac {dy^2 (t)}{dt^2} + \dfrac {3dy(t)}{dt} + 2y(t)=x(t)$ Calculate output y(t) if input $x(t) = e^{-3t} u(t) $ is applied to the system.

Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 10

Year : MAY 2014

By Laplace transform,

$X (s) = \dfrac 1{s+3}$

By Laplace transform,

$s^2y (s) +3s\space y (s) + 2 y (s) = x (s) \\ y (s)(s^2+3s+2) = x (s) \\ H (s) = \dfrac 1{(s^2+3s+2)} \\ =\dfrac 1{(s+1)(s+2) } \\ Y (s) = x (s) \times H (s) \\ =\dfrac 1{(s+3)(s+1)(s+2)} \\ Y (s) =\dfrac A{s+3}+ \dfrac B{s+1} + \dfrac C{s+2} $

By Inverse Laplace transform,

$y(t) = \dfrac 12 e^{-3t} u(t)+ \dfrac 12 e^{-t} u(t)-e^{-2t} u(t).$