Consider the system of differential equations with respect to time $t$:
$\dot{x}=y$
$\dot{y}=x-2x^3+y(x^2-x^4-y^2)$.
Let $L$ be the test function $\displaystyle L(x,y)=\frac{-x^2+x^4}{2}+\frac{y^2}{2}$. Answer the following questions( I'm inserting a pic because there are lot to type)
So I'm done for part a, but have no idea on c,d,e. At least hints for them are appreciated. For b, the curve which is identically zero is clearly a solution. For the second one I got that $y\dot{y}=\dot{x}[x-2x^3+y(x^2-x^4-y^2)], $ but could not derive the two differential equations separately. Can somebody please help?
EDIT: Note that $-1/8$ is the absolute minimum of $L$.