Assume we have a straight piece of wire with two end points and

and with length where and . The wire

has non-ohmic resistance and hence the current is not proportional

to the potential difference, i.e. . In

fact the current is a function of the voltage at and , that

is .

I know and hence I know the current. However, I do not know

as a function of . I tried several mathematical

tricks, mainly from the calculus of variation, trying to find

but I did not get a sensible result. Can any one suggest a method

(whether from the calculus of variation or other branches of mathematics)

to solve this problem and obtain .