This is a teaching-related question.
When teaching ordinary differential equations, we typically distinguish between the general solution (which includes unspecified integration constants) and particular solutions (where the integration constants are fixed by, say, initial values).
This makes sense for linear ordinary differential equations. But I can imagine that some nonlinear ordinary differential equations have two different solutions that look nothing like each other, that is, their difference is not an integration constant.
Do you know an example for such differential equations (and solutions)?
Best Answer
The classic counterexample is $y' = 2 \sqrt{|y|}$. It has "quadratic" solutions, looking like $y = (x-a)^2$ for $x \geq a$, but also $y = 0$ is a solution. You can patch these two together to obtain a nonunique solution to the initial value problem $y' = 2 \sqrt{|y|}$ and $y(0) = 0$.
It might also be fun to think about solutions to $y'= 2 \sqrt{|y|}$ and $y(0) = \eta$. Is the solution unique in that situation?