Differential equation defined only on $(-1,1)$

Question

We are going over differential equations, and I'm trying to think up of an example of a differential equation where the solution satisfying $x(0) = 0$ is defined for $t\in(-1,1)$, but I'm a bit stuck.

Any help appreciated!

Answer 1

An example differential equation satisfying the given conditions would be $$(1-t^2)x'(t)^2=1$$ whose solutions with $x(0)=0$ are $x(t)=\pm\sin^{-1}t$. The inverse sine, of course, is only defined on $(-1,1)$.

Answer 2

A piece of the tangent function comes to mind (compressed if you want it only defined on $(-1,1)$.)

So consider $y=\tan(\pi t/2)$. Then $$y'=\frac{\pi}{2}\sec^2(\pi t/2)$$ is already a differential equation, technically. You could make it autonomous since $\tan$ and $\sec$ have a Pythagorean relationship: $$y'=\frac{\pi}{2}\left(1+\tan^2(\pi t/2)\right)$$ $$y'=\frac{\pi}{2}\left(1+y^2\right)$$