Fancy View on a Simple Differential Equation
To fix ideas in my head, I made a picture which shows off the conceptual framework for solving differential equations with exterior differential systems for a very simple example: y' = x.
Of course, I used +The SageMathCloud
to generate the images. Since I made this, I might as well share it.
Notes on what is going on:
Think of a differential equation as a condition F(x,y,y') = 0 for some function F. Since F is a function of three variables, we can plot this level set! The key is to use z = y'. Then F(x,y,z) is no big deal. Our task will be to find a curve on this level set which represents the solution function y=y(x).
But we don't want to forget that z is really the y' direction. How does this matter? Well, if z = y' = dy/dx, that means that the allowed slopes are in that z direction. It becomes a problem to think about where to put that in the picture.
Following Cartan, the resolution is this:
z = dy/dx so...
z*dx - dy = 0.
Let w = z*dx - dy. This is a differential 1-form. At each point in space, it is a function that eats tangent vectors and spits out numbers. We
need a set-up where that is equal to zero... so at each point, we are only interested in the kernel of w. But that kernel is a plane!
If we collect all of these planes together, we get something called a distribution.
We can easily find a curve on the level set, but if we are going to do it in a way so that z = y', we must have that our curve is always tangent to the planes in this funny distribution.
To look closely, you can browse this SMC worksheet.