What precisely is a scalar autonomous differential equation? I'm confused about what this precisely means, more so because we did not discuss this in any lectures nor is it, as far as I can tell, defined in the notes. I've tried looking it up as well, but I couldn't find anything, only specific examples. It occurs in the following question:

Show that every scalar autonomous differential equation with $f\in\mathcal{C}^1$ is of gradient-type. Furthermore, show that the matrix of a linear gradient equation is symmetric.

Here gradient-type means that for a (system of) autonomous differential equation(s) $\dot{y}=f(y)$, there exists $V:\mathbb{R}^n\supset\!\to\mathbb{R}$ such that $f=\operatorname{grad}V$ with $V\in\mathcal{C}^2$.

Furthermore, what is a (linear) gradient equation? Do they simply mean a (linear) differential equation of gradient-type? The notes are not clear about it at all.

Any help would be appreciated. Thank you!


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