= Solution
Assume the <vector field> and multiplier are continuously differentiable, as required for <Green's theorem> and <uniqueness> of trajectories. A nonconstant periodic <trajectory> of a planar <autonomous system> is a simple closed orbit: <uniqueness> prevents crossings, and one traversal can be taken over its least period. Let $\Omega$ be its bounded interior. Along the orbit $dx=f\,dt$, $dy=g\,dt$, so
$$
\oint_{\partial\Omega}\phi f\,dy-\phi g\,dx=\int\phi(fg-gf)\,dt=0.
$$
<Green's theorem> identifies this with $\iint_\Omega[\partial_x(\phi f)+\partial_y(\phi g)]\,dx\,dy$, which is strictly positive by hypothesis, a contradiction. Thus there is \b[no nonconstant <periodic orbit>]. This standard periodic-orbit criterion does not by itself exclude stationary solutions.
For the first concrete system choose $\phi=-1$. Then
$$
\partial_x(\phi f)+\partial_y(\phi g)=e^{-x}>0.
$$
There are also no <equilibria>: $f=0$ requires $y=1$, but then $g=-e^{-x}-x^2<0$. Consequently it has \b[no <periodic solution>, including constant ones].
For the second system the origin is outside the domain of the printed <vector field>. For $r>0$,
$$
\dot r=\frac{x\dot x+y\dot y}{r}=1-r,\qquad\dot\theta=\frac{x\dot y-y\dot x}{r^2}=1.
$$
Thus for an admissible initial point with $\rho=\sqrt{a^2+b^2}>0$,
$$
\boxed{r(t)=1+(\rho-1)e^{-t},\qquad\theta(t)=\alpha+t,}
$$
where $a=\rho\cos\alpha$, $b=\rho\sin\alpha$. These formulas are valid while $r>0$, in particular for every $t\geq0$. Trajectories rotate counterclockwise, spiral outward when $0<r<1$, and inward when $r>1$, approaching the <unit circle>. On $r=1$ they give $x(t)=\cos(\alpha+t)$, $y(t)=\sin(\alpha+t)$, a \b[<periodic orbit> with least period $2\pi$]. It is an attracting <limit cycle>. There is no contradiction with the preceding criterion: this <vector field> is singular at the origin and has the exhibited closed orbit.
\Image[/past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2006-phase-portrait.png]
{title=Counterclockwise trajectories approach the attracting unit circle}
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