Okay the first thing we technically need to do here is apply separation of variables. Even though we did that in the previous section let’s recap here what we did.
First, we assume that the solution will take the form,
and we plug this into the partial differential equation and boundary conditions. We separate the equation to get a function of only t on one side and a function of only x on the other side and then introduce a separation constant. This leaves us with two ordinary differential equations