Solving Poisson's Equation in 1D: A Shortcut Using Variable Substitution

[th5418]

Homework Statement

Alright, well this is more of a math problem I guess... but here it is.
$$\frac{\partial^{2}}{\partial x{2}} V = - \frac{I}{A\epsilon_o} \sqrt{\frac{m}{2e}} \frac{1}{\sqrt{V}}$$

Everything besides the $$V$$ is constant.

Homework Equations

Trying to solve for the potential

The Attempt at a Solution

What's the general solution for these types of second order differentials?

 

[gabbagabbahey]

First, if it is a 1D problem, the partial derivatives can be replaced by ordinary derivatives!

Second, try defining a new variable: $W\equiv \frac{dV}{dx}$ and then take note of the fact that

[tex]\frac{d^2 V}{dx^2}=\frac{dW}{dx}=\frac{dV}{dx}\frac{dW}{dV}=W\frac{dW}{dV}=\frac{1}{2}\frac{d}{dV}(W^2)[/itex]

...you will then have a first order, separable ODE for $W$...which I'm sure you know how to solve!

P.S. This is a common trick, so it's worth remembering!