Mathematical Scratchpad

Detailed solution of RC Differential Equation

In class we were considering the following differential equation which arose from consideration of the time dependent charge on the capacitor of an RC circuit (the resistor and capacitor being situated in series. )

Some of the steps in the separation of variables were

Simplifying, we obtain

If we now plug in the initial condition Q( 0) = 0, we obtain

Now comes the fun part. Note the following:

1. Q is differentiable, so it must be a continuous function.

2. The right hand side of the above equation is never 0 because the exponential function always outputs strictly positive values.

3. Thus the function inside the absolute values, Q( t) -EC, is either always positive or always negative. (Because it is a continuous function that is never 0. )

4. Since, Q( 0) -EC = 0-EC < 0, we can conclude that Q(t) -EC is strictly negative for all t which allows us to replace | Q( t) -EC| with EC-Q( t) .

5. Thus, we can finally conclude

   | Q( t) -EC| = ECexp æ ç è - t RC ö ÷ ø EC-Q( t) = ECexp æ ç è - t RC ö ÷ ø Q( t) = EC-ECexp æ ç è - t RC ö ÷ ø = EC æ ç è 1-exp æ ç è - t RC ö ÷ ø ö ÷ ø

   exactly as claimed in class.