Step response of an RL Circuit

In calculating the step response of an RL circuit we consider the following circuit:

After the above switch is closed Kirchoffs' voltage law can be applied which gives:

Then rearranging the above we obtain the  following equation:

We now integrate each side and using x and y as variables for the integration we get:

thus integrating this gives:

where I₀ is the current at time t=0 and i(t) is the current at any time after 0. Taking inverse logs and rearranging gives us the following equation:

When the initial energy in the inductor is 0, I₀ is zero hence the above becomes:

Step Response of a RC circuit

If we consider the following circuit:

we can use a method similar to that of above to arrive at the expression for the voltage, which is: