Chapter 9.2, Problem 27E

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, C), so in this case Kirchhoff’s Law gives

$RI+\frac{Q}{C}=E(t)$

But I = dQ/dt, so we have

$R\frac{dQ}{dt}+\frac{1}{C}Q=E(t)$

Suppose the resistance is 5 Ω, the capacitance is 0.05 F, and a battery gives a constant voltage of 60 V.

1. (a) Draw a direction held for this differential equation.
2. (b) What is the limiting value of the charge?
3. (c) Is there an equilibrium solution?
4. (d) If the initial charge is Q(0) = 0 C, use the direction field to sketch the solution curve.
5. (e) If the initial charge is Q(0) = 0 C, use Euler’s method with step size 0.1 to estimate the charge after half a second.