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Consider the following autonomous first-order differential equation dy/dx = (y - 1)^4. Find the critical points and phase portrait of the given differential equation.

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Final answer:

To find the critical points of the differential equation dy/dx = (y - 1)^4, we set the derivative to zero, leading to the critical point y=1.

Step-by-step explanation:

The question involves analyzing the critical points of an autonomous first-order differential equation given by dy/dx = (y - 1)^4. To find the critical points of the differential equation, we set the derivative equal to zero. This happens only when y-1=0.

Thus, the critical point is y=1. At this point, the slope of the solution curves is zero, indicating we have a horizontal tangent. For the phase portrait, we observe the behavior of the solutions around the critical point. When y < 1, the derivative is positive, and when y > 1, the derivative remains positive.

This implies that the solutions will be increasing on both sides of the critical point; however, the increase is less steep as it approaches y=1 due to the fourth power in the derivative. The phase portrait would show a stable equilibrium at y=1 with arrows pointing towards the critical point.

User Leo Alekseyev
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