Does this differential equation have a solution?

1. xvudi

2
1. The problem statement, all variables and given/known data

I'm examining the equation Ohm_Max + dOhm/dr = Ohm_Max - dOhm/dr and can't find any solutions other than the trivial one, Ohm(r) = 0 for all r.

It's meant to determine if it is possible to build a length of conductor such that, upon dividing it at any arbitrary point, you'll find that the resistance behind is the same as the resistance ahead.

3. The attempt at a solution

Ohm(r) = 0. You can't have one because if Ohm(r) is even then dOhm/dr is odd.

Ohm(r) = Ohm_Max at r = 0 for bounds -L/2 to L/2, captured in the use of Ohm_Max the constant.

So the integral from -L/2 to 0 must equal the integral from 0 to L/2 meaning that dOhm/dr has to be even. This can't be as if Ohm(r) is odd then Ohm(0) must be 0 and not Ohm_Max. Odd functions cannot be valued at 0.

Now, mathematically why can't this work? I apologize for the absence of TeX. I'm still getting used to the forum interface.

Staff: Mentor

Instead of using Ohm as a variable, let's switch to R, for resistance. Your equation simplifies to 2 dR/dr = 0.

This is a very simple differential equation to solve, and the solution is not necessarily the trivial solution, although that is one solution.

3. xvudi

2
Ohm is resistance. I apologize for the confusion.

So R(r) = C1 * exp(0 * r) = C1

Particular solution for R_Max is 0 as all of its derivatives are zero, putting in our constraint at R(0) we solve for our constant C1.

R(0) = R_Max => C1 = R_Max except...

R(-L/2) = 0

R(L/2) = 0

So C1 must be a function of r except C1 cannot be a function of r.

This means that R does not depend on r. Meaning that dR/dr has to be zero (easily verified) and I just realized where my last post went off the rails.

I meant to add that for bounds -L/2 and L/2 that R(r) must be zero.

So the only way you can do it is if you stick a slider on some rails next to a circuit containing two perfectly matched resistors and call the resistors the system because I forgot the above constraint and I apologize.

So, with this added constraint, what is the grand mathgalactic reason why it isn't a solvable problem? It's something fundamental. It has to do with the problem type and I can't put my finger on it.

Staff: Mentor

This is really the long way around.
R'(r) = 0 ==> R(r) = C
The idea is that if the derivative of something is zero, the something must be a constant.

Now, using the initial condition R(0) = Rmax, then C = Rmax

If you also know that R(L/2) = R(-L/2) = 0, then C = Rmax = 0.

Am I missing something?