= Solution
{c}
Both components vanish at
$$
(0,0),\quad(1,0),\quad(0,1),\quad(4,3).
$$
The <Jacobian matrix> of the vector field is
$$
J(x,y)=\begin{pmatrix}y-2x+1&x\\-y&4y-x-2\end{pmatrix}.
$$
Its <eigenvalues> classify these <equilibria> by <linearization of a dynamical system>:
$$
\boxed{\begin{array}{c|c|c}
\text{point}&\text{eigenvalues}&\text{type}\\\hline
(0,0)&1,-2&\text{saddle}\\
(1,0)&-1,-3&\text{stable node}\\
(0,1)&2,2&\text{unstable improper node}\\
(4,3)&1-\sqrt{13},\ 1+\sqrt{13}&\text{saddle}
\end{array}}
$$
All real parts are nonzero, so the linear classifications determine the local stability of the nonlinear <autonomous system>.
At the origin, the stable direction is the $y$-axis and the unstable direction is the $x$-axis. At $(1,0)$ the eigendirections are $(1,0)$ for $-1$ and $(1,-2)$ for $-3$. At $(0,1)$,
$$
J=\begin{pmatrix}2&0\\-1&2\end{pmatrix}
$$
has only one eigendirection, the $y$-axis; its <linearization of a dynamical system> has solutions $\delta x=Ae^{2t}$, $\delta y=(B-At)e^{2t}$, explaining the <improper node>. At $(4,3)$, stable and unstable tangent vectors are respectively $(4,5-\sqrt{13})$ and $(4,5+\sqrt{13})$.
For the <phase portrait>, the axes are invariant, and the remaining <nullclines> are $y=x-1$ and $y=x/2+1$. Off these lines, the complete directional information is
$$
\operatorname{sgn}\dot x=\operatorname{sgn}(x)\operatorname{sgn}(y-x+1),\qquad
\operatorname{sgn}\dot y=\operatorname{sgn}(y)\operatorname{sgn}(2y-x-2).
$$
In the positive quadrant, arrows point right above $y=x-1$ and left below it, and point up above $y=x/2+1$ and down below it. On the positive $x$-axis, trajectories tend to $(1,0)$. On the $y$-axis, $0<y<1$ flows toward the origin and $y>1$ flows upward. These signs and the saddle directions give the following sketch; the plotted curves illustrate the <phase portrait>, with the marked saddle curves tangent to the calculated <eigenvectors>.
\Image[/past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2011-phase-plane.png]
{title=Phase portrait with the four equilibria, both non-axis nullclines, and stable and unstable separatrices of the saddle at (4,3). Arrows indicate increasing time.}
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