Given the solution to some differential equation, is the original equation necessarily unique?

Question

For example, if I have the fundamental solution set $\{x^2\}$, such that $y(x)=Cx^2$ is the solution to some unknown differential equation, is it guaranteed that only one such equation exists with this solution?

I know I can work backwards to show that this solution satisfies $\dfrac{dy}{dx}-\dfrac{2}{x}y=0$, so this might not be a great example... Are there any cases where there can be more than one differential equation corresponding to any given solution?

Answer 1

Not necessarily true. For example, $u'=u$ and $u'-xu = 1-x$ both are saisfied by the same family $u = Ae^x$.

Answer 2

In theory, there could be an infinite number of differential equations which have a particular function as a solution. Abel's formula would seem to indicate this. Off the top of my head, for $f(x)=x^2$, I can come up with $y''-2=0$ and $y(y'')-y'=0$.