Here is a short one about two simple differential equations. They both have “standard” solutions that appear in textbooks, but here is a method that treats them in similar ways.
First, an easy one. If is a (infinitely) differentiable function such that , then for some constant . Proof: Write . Then and hence is a constant function .
Here is another one. Let be a (infinitely) differentiable function defined on [ such that . Then for some constants and . First make the substitution , and set . Consider also the function . Then
Now we do the same for :
The numerator of the last term is actually a constant: its derivative is
which is 0 by the fact that .
Hence for some constant . Integrating twice, we get for some constants and . Now substituting back , we get . Since , we conclude that .