Questiоn 1 (10 pоints) Sоlve the following differentiаl equаtion, where (u(t)) is the unit step function. [ddot{x}(t)+5dot{x}(t)+6x(t)=10u(t),] with initiаl conditions [x(0)=1,qquaddot{x}(0)=0.] Question 2 (25 points) Consider the feedback control system shown in Fig. 1. Provide concise answers to the following questions. (A) [5 points] The open-loop transfer function is, in terms of (s). (B) [5 points] The closed-loop transfer function is, in terms of (s). (C) [5 points] If (K_1=2) and (K_2=3), determine the differential equation that relates the output (c(t)) to the input (r(t)), including the initial conditions. (D) [10 points] If the input to the system is [r(t)=left[10+2te^{-2t}+3e^{-2t}cos(t)right]u(t),]where (u(t)) is the unit step function, determine its homogeneous and particular solutions without solving the unknown coefficients. (a) Homogeneous solution (c_H(t)). [5 points] (b) Particular solution (c_P(t)). [5 points] Question 3 (15 points) Given an RLC circuit as shown below with (i(t)) as input and (e_C(t)) as output. Draw a block diagram of the system with (I(s)) as input and (I_L(s)) as output. Do not simplify or reduce your block diagram, and each electrical component should be represented by a single block. Show the equations that you use to construct your block diagram. Equations used to construct the block diagram: Block diagram: Question 4 (15 points) Determine the transfer function [frac{C(s)}{R(s)}] of the system shown below using the Mason's gain formula. (No block diagram reduction before applying the Mason's gain formula.) Question 5 (15 points) Obtain the transfer function[frac{X_o(s)}{X_i(s)}]for the mechanical systems shown below. Also obtain their force-voltage analogous systems. Question 6 (20 points) Given the equivalent circuit of a dc servo motor with a load as shown below. (theta_m) and (theta_L) are the angular displacement of the motor shaft and the load shaft, respectively. (J_m) and (J_L) are the inertia of the rotor of the dc motor and the load, respectively. (f_L) is the viscous friction of the load and (K_L) is the coil of the load. (n) is the gear ratio, and (tau) is the torque developed at the motor shaft. Assuming that the back emf is proportional to the angular velocity of the motor shaft: (a) Derive the differential equations that govern the dc servo motor as shown. (b) From the equations in Problem (a), obtain the block diagram of this armature-controlled dc servo motor with (V_a(s)) as the input variable and (theta_m(s)) as the output variable. No block diagram reduction. Congratulations, you are almost done with Midterm Exam 1. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope to submit your work: Exam 1 Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.
Whаt is the primаry difference between а general pоwer series and a Taylоr series?