How do we know that we found all solutions of a differential equation? I hope that's not an extremely stupid question, but it' been in my mind since I was taught how to solve differential equations in secondary school, and I've never been able to find an answer.
For example, take the simple differential equation $$y'=y.$$
While I understand that $y=C\cdot e^x$ satisfies this equation, I do not understand why there can't be other solutions as well. So, how do we know this?

In the case of y'=y...
Move along the x axis, follow the ordinate line up to the curve where the slope ='s the y-ordinate. In this way you trace out the curve, being only one curve save for translation.
