Question

Solve the differential equation V(t)=−CRdV(t)dt for the initial conditions given in the problem introduction to find the voltage as a function of time for any time t. Express your answer in terms of q0, C, R, and t.

Answer 1

Answer:

Explanation:

V(t)=−CRdV(t)/dt

(1/V(t))dV(t)=(−1/CR)dt

$\int\limits^v_{v0} {\frac{1}{v(t)} } \, dx = \int\limits^t_0 {\frac{-1}{CR} } \, dt \\\\ln (v)-ln(v_0)=\frac{-1}{CR}t \\\\ln (v/v_0)=\frac{-1}{CR}t \\\\v = v_0 e^{\frac{-1}{CR}t}$