RC series

A circuit of 2 component a resistor and a capacitor connected in series

${\displaystyle V_{C}+V_{R}=0}$

${\displaystyle C{\frac {d}{dt}}v(t)+{\frac {v}{R}}v(t)=0}$

${\displaystyle {\frac {d}{dt}}v(t)=-{\frac {1}{RC}}v(t)}$

${\displaystyle \int {\frac {dv(t)}{v(t)}}=-\int {\frac {1}{RC}}dt}$

${\displaystyle Lnv(t)=-{\frac {1}{RC}}t+c}$

${\displaystyle v(t)=e^{(-{\frac {1}{RC}}t+c)}}$

${\displaystyle v(t)=Ae^{-{\frac {1}{RC}}t}}$