Answer
See below.
Work Step by Step
a) The autonomous differential equation can be defined as a function of $y$ only, and can be expressed as: $K(y)=\dfrac{dy}{dx}$.
b) The autonomous differential equation can be defined as a function of $y$ only, and can be expressed as: $K(y)=\dfrac{dy}{dx}$.
When $K(y_0)=0$ for some value $y=y_0$, then this will represent a solution to the differential equation. These values are known as equilibrium values or equilibrium points.
c) The equilibrium values or equilibrium points do not involve any change in the dependent variables; this implies that $y$ wold remain stable.
d) Let $K(y)=\dfrac{dy}{dx}$ be an autonomous differential equation. Suppose $y(x)=a$ is at an equilibrium value; that is, $k(a)=0$. Then
i) If $\dfrac{da}{dx} \lt 0$, the equilibrium $y(x)=a$ is stable.
ii) If $\dfrac{da}{dx} \gt 0$, the equilibrium $y(x)=a$ is unstable.
