I've tried to solve a little problem that goes as follows:
Consider a system of ODEs: $x'=y-x^3\text{ and } y'=-x^3-y^3$ And the function $L(x,y)=\frac{1}{2}y^2+\frac{1}{4}x^4.$
Now I shall show that $(0,0)$ is not linearly stable, and that $L$ is a Lyapunov function.
I tried to investigate $\frac{dL}{dy}$ and $\frac{dL}{dx}$ and check if this is smaller or greater than $0$ at (0,0). However, these two terms are just $\frac{dL}{dy}=y,\frac{dL}{dx}=x^3$, so it seems we have to solve for $x,y$ explicitly? I'm confused because this looks like one of those exercise where brute-force can be avoided (And we did not deal with non-linear ODEs yet)?
yours, Marie