Final answer:
The causal LTI system with the given frequency response corresponds to the differential equation: y''(t) + 7y'(t) + 12y(t) = x'(t) + 5x(t).
Step-by-step explanation:
The causal LTI system mentioned in the question has a frequency response given by H(jω) = (jω + 5) / ((jω)² + 7(jω) + 12). This frequency response can be associated with a linear, constant-coefficient differential equation that describes the relationship between the system's input x(t) and output y(t).
To specify the differential equation for such a system, we match the frequency response to the transfer function of the differential equation. The denominator of the frequency response gives us the characteristic equation of the system which must match the differential equation coefficients. Similarly, the numerator indicates the relationship between the input and the derivative of the output.
The standard form of the second-order differential equation that corresponds to the given frequency response would be:
ay''(t) + by'(t) + cy(t) = dx(t)
Where ' denotes differentiation with respect to time t, and the coefficients a, b, c, and d are constant values determined by the terms in the given frequency response. Converting the jω terms back to time domain derivatives, we would equate a = 1, b = 7, c = 12, and d = 1. Therefore the linear differential equation would be:
y''(t) + 7y'(t) + 12y(t) = x'(t) + 5x(t).