Final answer:
The transfer or system function for the given differential equation is H(s) = (s^2 - 4s + 5) / (s^2 + 10s + 21), and the Region of Convergence (ROC) is s > -3, which is to the right of the rightmost pole at s = -3.
Step-by-step explanation:
To find the transfer or system function of the given differential equation, we first take the Laplace transform of both sides of the equation assuming that initial conditions are zero, as the system is causal. The Laplace transform of a derivative of a function is sY(s) - y(0) for the first derivative and s^2Y(s) - sy(0) - y'(0) for the second derivative, where Y(s) is the Laplace transform of y(t). Applying this to the differential equation, we have:
s^2Y(s) + 10sY(s) + 21Y(s) = s^2X(s) - 4sX(s) + 5X(s)
This can be rearranged to solve for the transfer function H(s), which is Y(s)/X(s):
H(s) = (s^2 - 4s + 5) / (s^2 + 10s + 21)
The Region of Convergence (ROC) for a causal system is typically to the right of the rightmost pole in the s-plane, since for a causal system all poles must be in the left half of the s-plane for the system to be stable.
To find the ROC, we need to determine the poles of the system function by setting the denominator equal to zero:
s^2 + 10s + 21 = 0
This factors to (s + 7)(s + 3) = 0, giving us poles at s = -7 and s = -3. Therefore, the ROC is s > -3.