1. Find the general solution of the second order differential equation y′′ - 2y′ + 4y = 541e2x cos(5x).
2. Consider an electric circuit consisting of an inductor with inductance L Henrys, a resistor with resistance R Ohms and a capacitor with capacitance C Farads, connected in series with a voltage source of V Volts. The charge q(t) Coulombs on the capacitor at time t ≥ 0 seconds satisfies the differential equation
L(d2q/dt2) + R(dq/dt) + q/C = V. Also, the current in the circuit i(t) Amps satisfies
i = dq/dt.
Suppose that in a particular circuit, L = 0.4 Henrys, R = 0 Ohms, C = 0.1 Farads and V = 110sin(ωt) Volts, where ω ∈ R. Initially the charge on the capacitor is 1 Coulomb and there is no current in the circuit.
(a) Write down the differential equation satisfied by q(t) in this circuit.
(b) Determine the value(s) of ω so that resonance occurs in the circuit.
(c) In the case where there is no resonance,
i. Solve the differential equation to find the charge on the capacitor at any time.
ii. Determine the transient and steady state solutions for the charge, if they exist.
iii. Find the current in the circuit at any time.
3. The electric potential energy v(r) of a charged particle located between two uniformly charged concentric spheres with radii r1 and r2 satisfies the second order differential equation
rv′′+2v′ = 0, r1 ≤ r ≤ r2
where r is the distance of the charged particle from the common centre of the spheres.
(a) Determine the general solution of the differential equation, by trying a solution of the form v(r) = rm, m ∈ R.
(b) Using your answer to part (a), find the electric potential energy of a charged particle between two concentric spheres with radii r1 = 2 cm and r2 = 20 cm, kept at potentials v1 = 220 Volts and v2 = 130 Volts respectively.