The solution of voltage along an RC transmission line as a function of distance and time
Consider the case of an infinite RC delay line:
If we isolate just a small part of the line, we can define currents and voltages:
If we take each element to represent a length, x, and rewrite voltages as a function of x, and rewrite each R and C as a capacitance per unit length, times a the length x, the equations can be rewritten as a partial differential equation
In the last line I wrote for simplicity (later).
Let us assume (without proof) that we can write v(x,t) as a product of two functions, one an equation of position alone (X(x)), and the other an equation of time alone (T(t)), so v(x,t)=X(x)T(t). This is a standard technique that works for a wide variety of partial differential equations.
Now on the left hand side we have something that is a function of position alone, and the right hand side is a function of time alone. For these equations to hold for any value of x and/or t, obviously they must both be constant. So we can choose an arbitrary constant, chosen to be -u2 for simplicity (later).
You have seen these equations before and they have solutions of the form
Continue on to the response of the line to a step input.