Question

In Problems 13 and 14 construct an autonomous first-order differential equation $\mathit{d}\mathit{t}\mathbf{/}\mathit{d}\mathit{x}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{y}\mathbf{\right)}$whose phase portrait is consistent with the given figure.

Short answer

The autonomous differential equation is $\frac{dy}{dx}={\left(y-1\right)}^2{\left(y-3\right)}^2$.

Step-by-step solution

Definition

An autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable.

Form differential equation

From the given phase portrait, we see that the differential equation has critical points 1 and 3.

Hence, f(y) should contain powers of (y-1) and (y-3). The following table explains the figure.

From the above information, since f(y) is always positive, it could be

$f\left(y\right)={\left(y-1\right)}^{2n}{\left(y-3\right)}^{2n}$

Therefore, one of the simplest functions can be for n = 1 and the corresponding differential equation whose phase portrait is given will be $\frac{dy}{dx}={\left(y-1\right)}^2{\left(y-3\right)}^2$.