how to find the solution for differential equation of an RC Circuit when the equation is: RC(dUc/dt)+Uc=E

Question

vincent-lyon.fr · Answer

There are different ways. Either you already know the general solution to such a first order linear equation, or you use separation of variables.

Put Uc on one side and time on the other side:

$\Large \frac{dU_c}{dt} =\frac {E-U_c}{RC}              $ , then

$\Large \frac{dU_c}{U_c-E} =-\frac {dt}{RC}              $

Integrate both sides from initial conditions (for instance t=0 and Uc=0) to any time and voltage t and Uc.

This develops to:

$\ln \Large \frac{U_c-E}{0-E} =-\frac {t-0}{RC}              $

Take exponential and rearrange to:

$U_c =E\: (1-\Large e\;\Large ^{ \frac {-t}{RC}})              $

anonymous · Answer

Thank's :) btw do we use the same method to find it for RL & RLC??

vincent-lyon.fr · Answer

For RL you can, but RLC needs help of mathematicians because it is a second order differential equation.

anonymous · Answer

ouh ok thx for your help :)