Answer to Question #268806 in Differential Equations for Nitish

Question

a capacitor C=0.01F in series with a resistor R=20 ohm is charged from a batter E=10V . assuming that initially the capacitor is completely uncharged , determine the charge Q(t) and the current i(t) at any time t.

Answer 1

$$RQ'+\dfrac{1}{C}Q=V$$

$$20Q'+\dfrac{1}{0.01}Q=10$$

$$Q'+5Q=0.5$$

$$Q'=-5(Q-0.1)$$

$$\dfrac{dQ}{Q-0.1}=-5dt$$

Integrate

$$Q=0.1+c_1e^{-5t}$$

Given $Q(0)=0$

$$0=0.1+c_1$$

$$c_1=-0.1$$

a)

$$Q(t)=0.1-0.1e^{-5t}$$

b)

$$i(t)=Q'(t)=0.5e^{-5t}$$