For a system of differential equations:
$$ x'=x^2-1,\\ y'=-y $$
(1)Draw the solution curve that starts at the intial point $Y(0)=(x(0),y(0))$.
(2)For the above solution curve $Y(t)=(x(t), y(t))$, draw the $(t, y(t))$ graph.
I know the answer of the first one, it's a straight line from origin the the point $(-1,0)$, but I am not sure what's the second question is about.
The picture is hard for me to see well, but based on what you say: if the solution $(x(t),y(t))$ looks like a horizontal line, what does that tell you about $(t,y(t))$?
You could also get the answer from observing that the system is decoupled and $y'=-y$. Once you've deduced what $y(0)=y_0$ is, the graph of $y(t)$ is determined.