Finding the limit of a function given the differential equation

Question

Unsure how to go about this question. I have written it out as: $y(x)=f(x)+2$ since $y(0)=2$, but I am assuming that $f(0)=2$. This doesn't really help me though as it is said that the differential equation must not be solved.

Answer 1

For small values of $x, y'(x) < 0$ Which is going to make y(x) a decreasing function. But as $y(x)$ approaches $1, y'(x)$ approaches $0.$ And what happens after that? $y(x)$ becomes stationary.

Answer 2

Let $f(y)$ be the right hand side of the ODE. Note that $f(y) <0$ for $y \in (1,7)$. Note that $1$ is an equilibrium (stable, but that does not matter). In particular, the solution must satisfy $y > 1$ for all $x \ge 0$.

Hence if we start at $y(0) = 2$ then $y$ will decrease and is bounded below by $1$, hence has a limit $y^*$. We must have $y^* = 1$, otherwise we have $y'(x) \le - \delta$ for some $\delta>0$ for all $x\ge 0$ which is a contradiction.