Answer
(b) $\frac{dy}{dt}=2+y$
Work Step by Step
The directional field for question 15 shows a field with an equilibrium solution at y=-2; it diverges as $t →∞$. Because there is only one equilibrium solution, it can be inferred that the differential equation is of the form $\frac{dy}{dt}=(a+by)^n$, this is because only equations in this form can have a single equilibrium solution. This rules out answers (d), (e), and (h). All remaining answer choices are of the form $\frac{dy}{dt}=a+by$ .
The equation is divergent, which means that $b\gt0$. This rules out answers (g), (i), and (j).
The differential equation has an equilibrium solution at $y=-2$, therefore $-\frac{a}{b}=-2$. So, $a=2b$. The equation is of the form $\frac{dy}{dt}=2b+by$, and b is positive. Therefore the answer is (b).
