Differential equations: how does separation of variables really work?

from blog Kevin Boone's website, | ↗ original
'Separation of variables' is one of the first methods usually taught to math and engineering students for solving differential equations, and yet the method as taught is mathematically very sloppy. That's not to say it doesn't work, but rather that it isn't made clear why it works, and whether similar methods will work elsewhere. This article...