Question

The response of the LTI system for \(\frac{d^2 y(t)}{dt^2} + \frac{dy(t)}{dt} + 5y(t) = \frac{dx(t)}{dt}\).  Given that y(0^–) = 2, \(\frac{dx(t)}{dt}\) (at t=0) = 0, x(t) = u(t) is __________

(a) 2e^-t cos t u(t)

(b) 0.5 e^-t sin t u(t)

(c) 2e^-t cos t u(t) + 0.5 e^-t sin t u(t)

(d) 0.5 e^-t cos t u(t-1) + 2e^-t sin t u(t-1)

I have been asked this question in a dream while sleeping

The above asked question is from Discrete Fourier Transform topic in chapter Fourier Transform of Signals and Systems

Answer 1

Correct answer is (c) 2e^-t cos t u(t) + 0.5 e^-t sin t u(t)

For explanation: s^2Y(s) – 2s + 2sY(s) – 2 + 5Y(s) = 1

∴ (s^2+2s+5) Y(s) = 3+2s

Or, Y(s) = \(\frac{2s+3}{s^2+2s+5}\)

= \(\frac{2(s+1)}{(s+1)^2 + 2^2} +  \frac{1}{(s+1)^2 + 2^2}\)

Hence, y (t) = 2e^-t cos t u(t) + 0.5 e^-t sin t u(t).