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Consider the following systems $x' = f(x,y), y' = g(x,y)$. Compute the expression in the (simpler) Bendixson-Dulac theorem ($\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}$) and decide if the theorem applies or not (if it applies, then the system has no closed trajectory). The region you take is the largest region in which both $f$ and $g$ are defined.

System: $\displaystyle x' = y-3 x^3, y'=x-y-7 y^3$.
$\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y} = {}$
Does the theorem apply: ? Theorem applies, sign positive, region simply connected Theorem applies, sign negative, region simply connected Theorem does not apply, problem: both positive and negative signs Theorem does not apply, problem: region not simply connected Theorem does not apply, problem: there are leprechauns in the equation None of the above

System: $\displaystyle x' = \frac{3 x}{\sqrt{x^2+y^2}}, y'=\frac{3 y}{\sqrt{x^2+y^2}}$.
$\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y} = {}$
Does the theorem apply: ? Theorem applies, sign positive, region simply connected Theorem applies, sign negative, region simply connected Theorem does not apply, problem: both positive and negative signs Theorem does not apply, problem: region not simply connected Theorem does not apply, problem: eels in the hovercraft None of the above

System: $\displaystyle x' = y, y' = 3(1-x^2)y - 3 x$.
$\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y} = {}$
Does the theorem apply: ? Theorem applies, sign positive, region simply connected Theorem applies, sign negative, region simply connected Theorem does not apply, problem: both positive and negative signs Theorem does not apply, problem: region not simply connected Theorem does not apply, problem: 42 None of the above

System: $\displaystyle x' = 3 e^x - 9 \cos(y), y'=3 e^y - 7 \sin(x)$.
$\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y} = {}$
Does the theorem apply: ? Theorem applies, sign positive, region simply connected Theorem applies, sign negative, region simply connected Theorem does not apply, problem: both positive and negative signs Theorem does not apply, problem: region not simply connected Theorem does not apply, problem: intersection of inequality quarks is normally distributed None of the above