The differential equation that can be set up as follows:

Given the values for R, L, C, and E, and noting that , we get the DE:

Let us solve the equivalent homogenous part first:

Assuming a solution of , we have the following characteristic equation:

.

Thus, the complementary solution is:

.

Now solve the non-homogeneous equation.

The technique I will use is known as the annihilator approach. I wasn't taught another way to solve these equations (besides Variation of Parameters, but I believe that it is unnecessary here).

First, we need to get the DE in Differential Operator Notation:

I will leave it for you (and others) to show that annihilates

Therefore, annihilates .

Applying the annihilator to both sides, we get:

Convert the equation so it has the form of the characteristic equation:

Thus the solution to the DE is:

where is the particular solution. The particular solution can't have arbitrary constants. To find and , which I will denote by and respectively, plug into the original DE.

You will find that

Therefore, the solution to the DE becomes:

Now apply the initial conditions .

Find :

.

Therefore,

coulombs

Woah...that was long. I hope you can absorb all of this!!