In the circuit below, R? = 1?, R? = 2?, and C? = C? = \frac{1}{2}F. The system is driven by the voltage source, v?(t).
C?
v?(t)
R? v?(t)
R?
v?(t)
C?
Complete the following parts to formulate the differential equation for this circuit and determine its solution using classical ODE methods.
(a) Write the KCL equations at the nodes defined by v?(t) and v?(t).
(b) Eliminate v?(t) from the equations derived in (a) to derive a single differential equation relating v?(t) to v?(t).
(c) Determine the characteristic polynomial and the natural frequencies of the differential equation found in (b).
(d) Determine the form of the homogeneous solution for t > 0.
(e) Find the particular solution for the input v?(t) = 2e?²? V, t ? 0.
(f) Assume that the capacitors are uncharged for t < 0 (i.e., the circuit was at rest) and determine and plot (by hand) the total solution v?(t) Vt.
