Determining the Stability of Equilibrium Solutions Draw a direction field for [latex]y^{\prime}(t)=2 y(t)[4-y(t)][/latex] and determine the stability of all equilibrium solutions.

We determined in example 3.6 that the equilibrium solutions are y = 0 and y = 4. We superimpose the horizontal lines y = 0 and y = 4 on the direction field in Figure 7.19. Observe that the behavior is distinctly different in each of three regions in this diagram: [latex]y>4,0<y<4 \text { and } y<0 \text {. }[/latex] We analyze each separately. First, observe that if [latex]y(t)>4 \text {, then } y^{\prime}(t)=2 y(t)[4-y(t)]<0 \text { (since } 2 y \text { is positive, but } 4-y \text { is negative). }[/latex] Next, if [latex]0<y(t)<4 \text {, then } y^{\prime}(t)=2 y(t)[4-y(t)]>0 \text { (since } 2 y \text { and } 4-y \text { are both }[/latex] positive in this case). Finally, if [latex]y(t)<0 \text {, then } y^{\prime}(t)=2 y(t)[4-y(t)]<0[/latex]. In Figure 7.19, the arrows on either side of the line [latex]y=4[/latex] all point toward [latex]y=4[/latex]. This indicates that[latex]y=4[/latex] is stable. By contrast, the arrows on either side of [latex]y=0[/latex] point away from [latex]y=0[/latex], indicating that [latex]y=0[/latex] is an unstable equilibrium.

Question 7.3.7: Determining the Stability of Equilibrium Solutions Draw a di...

Calculus: Early Transcendental Functions

Determining the Stability of Equilibrium Solutions

Draw a direction field for $y^{\prime}(t)=2 y(t)[4-y(t)]$ and determine the stability of all equilibrium solutions.

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We determined in example 3.6 that the equilibrium solutions are y = 0 and y = 4. We superimpose the horizontal lines y = 0 and y = 4 on the direction field in Figure 7.19.
Observe that the behavior is distinctly different in each of three regions in this diagram: $y>4,0<y<4 \text { and } y<0 \text {. }$ We analyze each separately. First, observe that if $y(t)>4 \text {, then } y^{\prime}(t)=2 y(t)[4-y(t)]<0 \text { (since } 2 y \text { is positive, but } 4-y \text { is negative). }$ Next, if $0<y(t)<4 \text {, then } y^{\prime}(t)=2 y(t)[4-y(t)]>0 \text { (since } 2 y \text { and } 4-y \text { are both }$ positive in this case). Finally, if $y(t)<0 \text {, then } y^{\prime}(t)=2 y(t)[4-y(t)]<0$ . In Figure 7.19, the arrows on either side of the line $y=4$ all point toward $y=4$ . This indicates that $y=4$ is stable. By contrast, the arrows on either side of $y=0$ point away from $y=0$ , indicating that $y=0$ is an unstable equilibrium.