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

\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]<br /> <br /> ...you will then have a first order, <b>separable</b> ODE for W...which I&#039;m sure you know how to solve!<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f609.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":wink:" title="Wink :wink:" data-smilie="2"data-shortname=":wink:" /> <br /> <br /> P.S. This is a common trick, so it&#039;s worth remembering!<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" />