Question

The system characterized by the differential equation \(\frac{d^2 y(t)}{t^2} –  \frac{dy}{dt} – 2y(t) = x(t)\) is _____________

(a) Linear and stable

(b) Linear and unstable

(c) Nonlinear and unstable

(d) Nonlinear and stable

This question was addressed to me at a job interview.

I need to ask this question from Fourier Analysis in chapter Fourier Analysis on Complex Spaces of Signals and Systems

Answer 1

Right option is (b) Linear and unstable

To elaborate: \(\frac{d^2 y(t)}{t^2} –  \frac{dy}{dt} – 2y(t) = x(t)\)

Now, x (t) –> h (t) so, the system is linear.

Again, taking Laplace Transform with zero initial conditions, we get, s^2 Y(s) – s Y(s) – 2Y(s) = X(s)

Or, H(s) = \(\frac{Y(s)}{X(s)} = \frac{1}{s^2-s-2} = \frac{1}{(s-2)(s+1)} \)

Since pole is at s = +2, the system is unstable.