(Solved): Problem 1 - RC Circuit Analysis w/ Differential Equations [10 Points \( ] \) In this problem, stud ...

Problem 1 - RC Circuit Analysis w/ Differential Equations [10 Points \( ] \) In this problem, students apply two different circuit analysis techniques to derive the differential equation for the step-response of the state variable using the same circuit. In future courses, like EDL ECEN 2270, students will derive the state-variable response to analog circuits which will automatically control robots based on input voltages/currents from sensors using the same type of analysis used in this problem. The capacitor is initially discharged. Due to the cumulative nature of the problem, no credit will be awarded for \( \mathrm{c} \) ) if either/both \( \mathrm{a} \) ) and \( \mathrm{b} \) ) are missing, incorrect, or incomplete. a) [4 pts] Using Combined Constraints Method, derive the differential equation for \( v_{c}(t) \) in terms of \( R, C, u(t) \), and any scalar constants in simplified standard form. b) [4 pts] Derive the differential equation for \( v_{c}(t) \) in terms of \( R, C, u(t) \), and any scalar constants using only Mesh Analysis Method in simplified standard form. c) [2 pts \( ] \) Find the general solution to the first-order differential equation from a) \& b) using only the Method of Undetermined Coefficients to find \( v_{c}(t) \) for \( t \geq 0 s \), the state-variable response, in terms of \( V_{s}, \mathrm{R}, \mathrm{C} \), and any scalar as a simplified symbolic function of time for \( t \geq 0 \mathrm{~s} \).