The following diff. equation describes the functionality of a system with respect to time. However, it is not known how the system will behave when stimulated. Apply a forward Laplace transform to determine damping ratio and pole zeros. Plot a pole zero diagram and comment on stability. d2y/dt2 + 6dy/dt + 10y = 63dy/dt + 63x Putting like terms on either side of the equation: d2y/dt2 + 6dy/dt - 63dy/dt + 10y = 63x = d2y/dt2 - 57dy/dt + 10y = 63x Laplace transform: s2Y(s) - 57sY(s) + 10Y(s) = 63 .... (63 because unit impulse used to stimulate?) Simplify for Y(s): Y(s).(s2 - 57s + 10) = 63 Solve for Y(s): Y(s) = 63/(s2 - 57s + 10) Using quadratic formula to find poles: s= (57 +/- √572 - 4 x 1 x 10)/(2 x 1) = (57 +/- 56.648)/2 = 28.5 +/- 56.648 (I was expecting a complex number!) I think that I am nearly there, but I suspect that I have gone wrong.