Question

Write down a differential equation of the form dy/dt = ay + b whose solutions have the required behavior as t → ∞. all other solutions diverge from y = 2.

Answer 1

The ODE has an equilibrium point at

$\dfrac{\mathrm dy}{\mathrm dt}=ay+b=0\implies y=-\dfrac ba$

We want $y=2$ to be an unstable equilibrium solution - in other words, we want all solutions with initial condition $y(t_0)=c$ for $c$ near 2 to diverge from $y=2$ - so $-dfrac ba=2$.

Divergence from $y=2$ would require that for $y>2$, any solution is increasing, and for $y$, and solution is decreasing. In order for that to happen, we need

$\dfrac{\mathrm dy}{\mathrm dt}=ay+b>0\implies y>-\dfrac ba$

if $y>2$, and

$\dfrac{\mathrm dy}{\mathrm dt}=ay+b$

if $y$.

Now we just pick $a,b$ to satisfy these conditions. An easy choice is $a=1$, which forces $b=-2$, so that one possible ODE would be

$\dfrac{\mathrm dy}{\mathrm dt}=y-2$

I've attached a plot of the slope field to demonstrate the behavior of the solutions.