Question

Problem 3: Insights into Differential Equations a. Consider the differential equation 습 +4 = f(t), where f(t) = e-u, 12 0. Please write the forms of the natural and forced solution for this differential equation. You DO NOT need to solve. (7 points) b. Again consider the differential equation f(t), where f(t) is an input and y(t) is the output (response) of interest. Please write the differential equation in state-space form. (10 points) c. The classical method for solving differential equations (also known as the method of un determined coefficients) has four steps. In no particular order, these steps are to: 1) find the coefficients of the natural solution by matching the total solution with initial conditions, 2) writing down the form of the forced solution, 3) writing down the form of the natural solu- tion, and 4) finding the coefficients of the forced solution by plugging it into the differential equation. Please list the four steps in the proper order. (6 points) d. You can drive an unknown linear circuit with any forcing function (input) f(t), and obtain its zero-state response (output) yzs(t). Applying the forcing function (input) f (t)- sin(2t), 12 0, you find experimentally that the zero-state response is yzs(t) sin(2t)-2cos (2t) + 20. Please find the sinusoidal steady-state response, when the forcing function is f(t)= cos(2t). (8 points)

Answer 1

4匕 氿tle lb