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.

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

From the Question we are told that

Generally,

Where

Therefore

Substituting

Then all other solutions diverge from

Therefore

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The ODE has an equilibrium point at

We want to be an unstable equilibrium solution - in other words, we want all solutions with initial condition for near 2 to diverge from - so .

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

if , and

if .

Now we just pick to satisfy these conditions. An easy choice is , which forces , so that one possible ODE would be

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

b

2

−

b

b

3

+

7

b

2

+

14

b

+

6

%

−

3

b

3

+

7

b

2

+

14

b

+

6

%

step-by-step explanation: