Laplace's Equation and Seperation of Multivariable Differential Equation

[dduardo]

Can someone explain how to separate a multivariable differential equation into two independent differential equations? I'm having an issue solving for the potential in spherical co-ordinates in terms of r and theta.

 

[BLaH!]

Let's say you have a differential equation that involves both $$r$$ and $$\theta$$. The way separation of variables works is that you assume that the solution can be written as a product of two functions: one function that is ONLY a function of $$r$$ and another function that is ONLY a function of $$\theta$$. Thus, you guess the solution

$$u(r, \theta) = R(r)\cdot \Theta(\theta)$$

You then insert this into your differential equation. Derivatives with respect to $$r$$ will only affect the $$R(r)$$ function (the $$\Theta(\theta)$$ is not affected) and vice versa with derivatives with respect to $$\theta$$.

After you have determined how your differential equation looks when you have inserted this solution, you can rearrange the differential equation so that it looks something like

$$F(R''(r), R'(r), r) = G(\Theta''(\theta), \Theta'(\theta), \theta)$$ (1)

Here I have assumed that your differential equation was second order in both $$r$$ and $$\theta$$. To put it in this form usually requires dividing both sides of the equation by $$R(r)\cdot \Theta(\theta)$$ but it can also involves other kinds of multiplications as well. The whole point of writing as I did in (1) is so you have one side of the differential equation that is completely independent of $$\theta$$ and the other side that is completely independent of $$r$$.

When this is true, the only possibility is that each side of the differential equation is equal to constant. This is true because the ONLY function of say $$\theta$$ that does not depend on $$\theta$$ is a constant.

At this point you are done...you now have the following:

$$F(R''(r), R'(r), r) = constant = G(\Theta''(\theta), \Theta'(\theta), \theta)$$

So you have TWO differential equations of a single variable. In this process however you must realize that separation of variables only works in some cases and in the cases where it does work you must ask the question "Is this solution unique?" But for the problems you are doing in E&M you probably don't have to worry about this.

 

[dduardo]

Thanks for the help. After I got both differential equations I was able to apply cauchy-euler to solve them.