Studying for my midterm, would appreciate if someone could help me break this down piece by piece

Problem 3 (Circuit Tuning). Recall that a forced LRC circuit (i.e. a circuit consisting of a inductor, a resistor, and a capacitor connected in series, together with a voltage source) is described by the differential equation LI" + RI' + 2 1 = E'(t), where E(t) is the function that describes the input voltage. It often makes sense to assume the voltage source is sinusoidal; in this problem, we will take E(t) = sin wt. Unless the circuit is damped (i.e., R = 0), it is impossible for the circuit to undergo true resonance. However, under certain conditions there is a resonant frequency w* that will maximize the amplitude of the steady-state response. This phenomenon gives us a practical form of resonance for damped systems. (a) Suppose we take L = 0.012 henrys, R = 70 ohms, and C = 0.000004 farads. Determine the steady-state response for this circuit. (b) Compute the amplitude of the steady-state response in terms of the frequency w. (c) Find the value of w that maximizes the amplitude of the steady-state response. In practice, there are situations where we might like to "tune" the circuit to a certain resonant frequency by adjusting the properties of the circuit. We can achieve such a result by installing a variable capacitor, which allows us to set the capacitance-hence the resonant frequency-to many different values. This sort of setup is useful for tuning a radio-if we would like to tune the radio to a particular frequency, we will need the circuit to experience resonance at that frequency. (d) Suppose we want to tune in to WWSC at 1450 AM. This station broadcasts at a frequency of 1450 kHz, so we need to take w = (1450) (1000) (27) = 1450000m. Assuming we keep the same values of L and R as in part (a), find the steady-state response in terms of the capacitance C. (e) Compute the amplitude of the steady-state response in terms of C. (f) Find the value of C that maximizes the amplitude of the steady-state response. This is the capacitance that is needed to tune our radio to the desired station