>>11297871I believe they go the idea for the Laplace Transform from the Fourier Transform. They hammered it out in their math rooms and figured out that it actually works. It's an interesting technique.
Anyway, how would you solve this?
Where A, B, and C are arbitrary constants?
Spoiler alert, you basically just plug in exponentials with a certain power (like ) and then solve for your 's.
There is some more subtlety than that and you should read up on it, but that's the general idea.
As for nonlinear differential equations (which would involve terms like y''^2) there are sometimes ways to get solutions, but usually not. These are generally classified "unsolvable", although there are ways to approximate solutions in certain domains. Getting these approximations is often part of the task of a physicist or engineer.
Ok, so how does this help us with the Heat Equation? This is a PDE
On the left we have a term which is a derivative with respect to space, on the right we have a term which is a derivative with respect to time. We simply assume they equal a constant (there are good reasons to make this assumption), and then solve the two separately. And we get sines and cosines out.
You probably won't understand what I've been saying, and that's because you need to learn about techniques to solve Differential Equations. That will lead you to Fourier series naturally
