Question

Find the transfer function of the system with differential equation \(\frac{{{d^2}y}}{{d{t^2}}} + 6\frac{{dy}}{{dt}} + 25y = 9u + 3\frac{{du}}{{dt}}\)
1. \(\frac{{U(s)}}{{Y(s)}} = \frac{{(3s + 9)}}{{({s^2} + 6s + 25)}}\)
2. \(\frac{{Y(s)}}{{U(s)}} = \frac{{(9s + 3)}}{{({s^2} + 6s + 25)}}\)
3. \(\frac{{Y(s)}}{{U(s)}} = \frac{{(3s + 9)}}{{({s^2} + 6s + 25)}}\)
4. \(\frac{{Y(s)}}{{U(s)}} = \frac{{(9s + 3)}}{{({s^2} + 25s + 6)}}\)

Answer 1

Correct Answer - Option 3 : \(\frac{{Y(s)}}{{U(s)}} = \frac{{(3s + 9)}}{{({s^2} + 6s + 25)}}\)

Analysis:

\(\frac{{{d^2}y}}{{d{t^2}}} + 6\frac{{dy}}{{dt}} + 25y = 9u + 3\frac{{du}}{{dt}}\)

Taking laplace transform on both sides,

s²Y(s) + 6sY(s) + 25Y(s) = 9U(s) + 3sU(s)

Y(s) (s² + 6s + 25) = U(s) (9 + 3s)

\(\frac{{Y(s)}}{{U(s)}} = \frac{{(3s + 9)}}{{({s^2} + 6s + 25)}}\)