2. We can determine the differential equation relating y(t) and a(t)from the system function found in(a). Since H(s)= xo, we multiply the denominator of H(s) by y(s) and we multiply the numerator of H (s by X(s) Y(s)(s2+3s+2)=X(s)(s+3) Distributing on both sides s Y(s+3sY(s+2Y(s=sX(s)+3X(s) Because of linearity, we can take the inverse Laplace transform of each term above and get the differential equation y(t) (t) 3r(t) Problem 4 Suppose we are given the following information about a causal and stable LTI system ith impulse response h(t)and a rational function H (s) The steady state response to a unit step, i.e., s(oo)=3 When the input is eu(t), the output is absolutely integrable−1 × √e →m 2. We can determine the differential equation relating y(t) and x(t) from the system function found in (a). Since H(s) = Y (s) X(s) , we multiply the denominator of H(s) by Y (s) and we multiply the numerator of H(s) by X(s): Y (s)(s2 + 3s + 2) = X(s)(s + 3). Distributing on both sides: s2Y (s) + 3sY (s) + 2Y (s) = sX(s) + 3X(s) Because of linearity, we can take the inverse Laplace transform of each term above and get the differential equation: d2y(t) dt2 + 3dy(t) dt + 2y(t) = dx(t) dt + 3x(t). Problem 4 Suppose we are given the following information about a causal and stable LTI system with impulse response h(t) and a rational function H(s): • The steady state response to a unit step, i.e., s(∼) = 1 3 . • When the input is et u(t), the output is absolutely integrable. 8