Assume that <-- Question: Shouldn't it be ?
The following process is standard for separable functions:
Now insert these into your differential equation:
Now divide both sides of this by the product RT:
Typically we would need to isolate the functions R and T to opposite sides of the equation, but this is already in that form.
Note that the LHS is only dependent on and the RHS only depends on t. Thus each side must be equal to the same constant. I'll call it , where is positive. (It might be more convenient to call it something else. This is my "gut" instinct on what might be useful.)
Thus your differential equations are: