Steady state and boundary conditions

To make life easier, we set E=0. The solution to (12) depends on the initial spatial distribution of voltage and the conditions at the ends of the cable, the boundary conditions. We will consider either a finite cable of length l or a semi-infinite cable where one end goes to infinity. We will first look only at the steady state behavior of the cable. In this case, we drop the derivative with respect to t and obtain:  

$$0 = \lambda^2 \frac{d^2V}{dx^2} - V$$ (12)

whose general solution is one of the following forms:

We define the input resistance of a cable as the ratio of the steady state potential divided by the current injected.