First Order Transient Circuits

first order circuit are those circuits whose response can be expressed by a first order differential equations.

Example of such circuits include the RL and RC circuits

RL circuit : circuit that contains an indicator and a resistor.

RC circuit : circuit that contains a capacitor and resistor

This is a first order linear ordinary non-homogenous differential
equation describing the response of the capacitor voltage.
It is first order because the highest degree of derivative is one.
it is called linear because the differential equation is linear
function of (dvc /dt) and vc(t)

To solve this circuit I'm going to use the differential equation approach. I'm concerned with the

voltage across the capacitor at and


At position 1 before the switch is thrown

At position 2 after switch is thrown I have the following from KCL:



This reduces to:

My problem is that I know that this is correct but I don't know how to put it all together.

I know that the solution is of the form:



The answer is

How do I extract this from my work?