When working with qualitative properties of ODE, its important to still be able to write rigorous proofs. I would like to better understand how to write a rigorous proof that is still clear and succinct for qualitative properties of an ODE. As a concrete case which can be used:
Consider the ODE $$y' = y^2 - x^2.$$ Suppose a solution has an extremum at $x_0 = -1$. Is it a maximum or minimum?
Source: MIT 18.03SC
My proof was:
It is a minimum. Since $y'$ is defined everywhere, it must vanish at the extrema, giving $$y'_0 = 0 \\ y_0 = \pm 1.$$
Furthermore, for sufficiently small $h$, then $$\begin{align*} y'(x_0+h) &= y_0' +hy_0'' + o(h) \\ &= h(2y_0y' - 2x_0) + o(h) \\ &= 2h + o(h) \end{align*}$$ proving that $x_0$ is a minimum.
Yet the supplied solution seems to be much simpler:
Extrema occur when $y = 0$; that is, along the nullcline. $y'' = 2yy' − 2x$, which is $−2x$ on the nullcline. At $x = −1$ this is positive, so we have a minimum.
I clearly had the key insight, but my proof, although correct, is cumbersome. What methods can be used to zero-in like the provided proof does and bypass the cobwebs?
My hunch is that my proof is cumbersome because it fails to leverage existing theorems: namely, that the second derivative at an extremum can show whether it is a minimum or maximum. Instead, I went back to the definition of minimum or maximum, which requires reasoning about the entire interval. Once you bring in the entire interval, writing an exact equation is difficulty, because we only know properties in the limit. In contrast, the provided proof only needs to talk about a single point - the extremum itself - which allows its reasoning to be much simpler.