The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (Ω ).
The voltage drop across the capacitor is Q/C, where Q is the charge (in coulombs), so in this case Kirchhoff€™s Law gives RI + Q/C = E(t) But l = dQ/dt, so we have R dQ/dt + 1/C Q = E(t) Suppose the resistance is SΩ, the capacitance is 0.05 F, and a battery gives a constant voltage of 60 V.
(a) Draw a direction field for this differential equation.
(b) What is the limiting value of the charge?
(c) Is there an equilibrium solution?
(d) If the initial charge is Q (0) = 0C, use the direction field to sketch the solution curve.
(e) If the initial charge is Q (0) = 0C, use Euler€™s method with step size 0.1 to estimate the charge after half a second.