We are given the differential equation:

$y' = (y - 3)(y^2 + y - 2)$

Step 1: Find the Equilibria

Equilibria occur when $y' = 0$, meaning we need to solve:

$(y - 3)(y^2 + y - 2) = 0$

Factor the quadratic:

The quadratic $y^2 + y - 2$ factors as:

$y^2 + y - 2 = (y - 1)(y + 2)$

Thus, the equation becomes:

$(y - 3)(y - 1)(y + 2) = 0$

The equilibria points are the roots of this equation:

$y = 3, \quad y = 1, \quad y = -2$

Step 2: Classify the Equilibria (Sinks, Sources, or Nodes)

To classify the equilibria, we need to analyze the sign of $y'$ in the intervals between the equilibria points.

The critical points divide the $y$-axis into four intervals:

1. $(-\infty, -2)$
2. $(-2, 1)$
3. $(1, 3)$
4. $(3, \infty)$

We will pick a test point in each interval and check the sign of $y'$.

• Interval $(-\infty, -2)$: Test $y = -3$
  $y' = (-3 - 3)(-3 - 1)(-3 + 2) = (-6)(-4)(-1) = -24$ Since $y' < 0$, the solution is decreasing in this interval.

• Interval $(-2, 1)$: Test $y = 0$
  $y' = (0 - 3)(0 - 1)(0 + 2) = (-3)(-1)(2) = 6$ Since $y' > 0$, the solution is increasing in this interval.

• Interval $(1, 3)$: Test $y = 2$
  $y' = (2 - 3)(2 - 1)(2 + 2) = (-1)(1)(4) = -4$ Since $y' < 0$, the solution is decreasing in this interval.

• Interval $(3, \infty)$: Test $y = 4$
  $y' = (4 - 3)(4 - 1)(4 + 2) = (1)(3)(6) = 18$ Since $y' > 0$, the solution is increasing in this interval.

Equilibrium Classification:

• $y = -2$: Source (since the solution moves away from it, $\uparrow \downarrow$)
• $y = 1$: Sink (since the solution moves toward it, $\downarrow \uparrow$)
• $y = 3$: Source (since the solution moves away from it, $\uparrow \downarrow$)

Step 3: Vertical Phase Line Representation

We represent the phase line with equilibria at $y = -2$, $y = 1$, and $y = 3$, using arrows to indicate the direction of the solution.

Here’s the vertical phase line:

& \quad \uparrow \\ & y = 3 \ (\text{Source}) \\ & \quad \downarrow \\ & \uparrow \\ & y = 1 \ (\text{Sink}) \\ & \downarrow \\ & \uparrow \\ & y = -2 \ (\text{Source}) \\ & \downarrow \\ \end{aligned}$$ ### Summary of Results: - $$y = 3$$: Source (solution moves away from this point) - $$y = 1$$: Sink (solution moves toward this point) - $$y = -2$$: Source (solution moves away from this point) --- Would you like to dive into any particular detail of this? Here are 5 related questions: 1. How do you determine if a point is a source or sink mathematically? 2. What happens at an equilibrium point that is neither a source nor a sink? 3. Can a differential equation have more than one sink or source? 4. How does the behavior of the solution change near a saddle point compared to a sink or source? 5. Can you represent this phase line for a more complex system of equations? **Tip:** To classify equilibria, checking the sign of the derivative $$y'$$ in intervals around the equilibrium is crucial for determining the behavior (sink or source).