Question

In Problems 39each 40 figure represents a portion of a direction field of an autonomous first-order differential equation $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}\mathbf{=}\mathit{f}\mathbf{\left(}\mathit{y}\mathbf{\right)}$. Reproduce the figure on a separate piece of paper and then complete the direction field over the grid. The points of the grid are $\mathbf{(}\mathit{m}\mathit{h}\mathbf{,}\mathit{n}\mathit{h}\mathbf{)}$, where $\mathit{h}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathit{m}$and $\mathit{n}$integers, $\mathbf{-}\mathbf{7}\mathbf{⩽}\mathit{m}\mathbf{⩽}\mathbf{7}\mathbf{,}\mathbf{-}\mathbf{7}\mathbf{⩽}\mathit{n}\mathbf{⩽}\mathbf{7}$. In each direction field, sketch by hand an approximate solution curve that passes through each of the solid points shown in red. Discuss: Does it appear that the DE possesses critical points in the interval $\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{⩽}\mathit{y}\mathbf{⩽}\mathbf{3}\mathbf{.}\mathbf{5}$? If so, classify the critical points as asymptotically stable, unstable, or semi-stable.

Short answer

Point$y=2$ is unstable &$y=-2$ is stable

Step-by-step solution

Definition

An equilibrium solution is said to be Asymptotically Stable if on both sides of this equilibrium solution, there exists other solutions which approach this equilibrium solution. An equilibrium solution is said to be Semi-Stable if one side of this equilibrium solution there exists other solutions which approach this equilibrium solution, and on the other side of the equilibrium solution other solutions diverge from this equilibrium solution. An equilibrium solution is said to be Unstable if on both sides of this equilibrium solution other solutions diverge from this equilibrium solution.

Figure

We are given the differential equation is autonomous, therefore the direction for all x and a fixed y are the same.

The curves show critical point at $y=\pm 2$. The critical point$y=2$ is unstable and the critical point $y=-2$is stable.