How do I solve this?

Consider the circuit from P2 in PS2, now connected to an external source as shown in figure . The external source is an AC source with VAC(t) = Vo cos(It + 0). R, L, Vo, $2, o are arbitrary positive constants. B. R 0000 Figure 2: 'R+LC' circuit with an external source. (a) Find the differential equation that describes the current in the inductor. You can use all what you learned about this circuit in PS2 but show again your work, including Kirchhoff equations. Remember to have sum of potentials to set the differential equation equal to V(t). Hint: just the differential equation. Do not solve it yet. (5pts) (b) Find IL(t) ignoring the transient behavior. That is, get the steady response to the external sourcell. Hint: one way is to set In(t) = Acost + B sin It, with A, B some constants. Then, find A, B by solving the differential equation of IL(t). Some suggestions from class: i/ do not use a coefficient of 1 in the I term. Use instead (RLC) I. The algebra will have less fractions. ii/ Beware that V(t) has an arbitrary phase: o. Use cos(St + 6) = costcoso - sin St sin o. iii/ at the end, check that A, B have units of current! (8 pts) (c) Given In(t), find the voltage across the coil inductor VL(t) ? Leave the answer in terms of A, B, or whichever labels you used in the previous questions. That is, you can, but it is not required to substitute the explicit expressions of A, B found in the previous question. (2 pts) (d) Find an expression for the marimum current in the inductor over one cycle of the external AC source: IL,max. Does it depend on o? Hints: i/ you do not need to take any derivatives to answer this question. ii/ make sure that IL, max has units of current. (3 pts) (e) Find the value of the external frequency, $2 > 0, that marimizes IL, max while keeping Vo, R, L, C constant. Let's call it max. What's the relationship among R, L, C that allows the circuit have an oscillatory response? Suggestion: check that the units of max are 1/s. (4 pts) (f) An LC circuit (i.e. one with R = 0) shows SHO at an angular frequency, called the resonant frequency, ORes = 1/VLC. Find the ratio max/Res. If we hold The phase of the external source o may be negative, but it can always be defined within [0, 2x) without loss of generality.L, C constant and vary R, do we need to increase or decrease R in order to get max closer to Res ? (3 pts)