6. Consider the differential equation $$ \frac{d y}{d t}=y(y-1)^{2}(y-2)^{3}(y-3)^{4} $$ Select the equilibrium point(s) in the list below that are nodes: $$ y=0, \quad y=1, \quad y=2, \quad y=3 $$

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In the given differential equation: $$ \frac{d y}{d t} = y(y-1)^{2}(y-2)^{3}(y-3)^{4} $$ the equilibrium points occur where $$\frac{dy}{dt} = 0$$, which are $$y = 0$$, $$y = 1$$, $$y = 2$$, and $$y = 3$$. ### Understanding the Nature of Each Equilibrium Point 1. **$$y = 0$$ (Multiplicity 1):** - **Behavior:** - For $$y < 0$$, $$\frac{dy}{dt} > 0$$. - For $$y > 0$$, $$\frac{dy}{dt} > 0$$. - **Interpretation:** The solution does not cross $$y = 0$$ and flows away from it on both sides, indicating it's a **repelling equilibrium**. 2. **$$y = 1$$ (Multiplicity 2):** - **Behavior:** - For $$y < 1$$, $$\frac{dy}{dt} > 0$$. - For $$y > 1$$, $$\frac{dy}{dt} > 0$$. - **Interpretation:** The solution does not cross $$y = 1$$ and flows away on both sides, making it a **non-crossing equilibrium**. Due to its even multiplicity, it behaves like a **node**. 3. **$$y = 2$$ (Multiplicity 3):** - **Behavior:** - For $$y < 2$$, $$\frac{dy}{dt} < 0$$. - For $$y > 2$$, $$\frac{dy}{dt} > 0$$. - **Interpretation:** The solution crosses $$y = 2$$, changing direction from decreasing to increasing. This indicates a **semi-stable equilibrium**. 4. **$$y = 3$$ (Multiplicity 4):** - **Behavior:** - For $$y < 3$$, $$\frac{dy}{dt} > 0$$. - For $$y > 3$$, $$\frac{dy}{dt} > 0$$. - **Interpretation:** Similar to $$y = 1$$, the solution does not cross $$y = 3$$ and flows away on both sides, behaving like a **node** due to its even multiplicity. ### Conclusion **Equilibrium points with even multiplicities (i.e., $$y = 1$$ and $$y = 3$$) behave as nodes** in the phase line analysis of this differential equation. **Answer:** Both y = 1 and y = 3 are equilibrium points that are nodes. Both $$ y = 1 $$ and $$ y = 3 $$ are equilibrium points that are nodes.

Consider the differential equation

Select the equilibrium point(s) in the list below that are nodes:

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Understanding the Nature of Each Equilibrium Point

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Consider the differential equation Select the equilibrium point(s) in the list below that are nodes: