91Ó°ÊÓ

Finding critical points in a differential equation involves determining where the derivative equals zero. In the equation \( \frac{dx}{dt} = (x+2)(x-2)^2 \), critical points occur when \( \frac{dx}{dt} = 0 \).
To solve this, you would set each factor in the function equal to zero. Hence, solving \( x+2 = 0 \) gives \( x = -2 \), and solving \( (x-2)^2 = 0 \) gives \( x = 2 \). These points are crucial as they indicate where the function's growth changes direction either stabilizing, destabilizing, or neither.
To summarize the process:

• Set the differential equation to zero.
• Solve the resulting algebraic equations.
• Identify solutions as your critical points.

The critical points tell us a lot about the behavior of the differential equation under small changes.

Stability analysis examines how small changes around critical points influence the system. For the differential equation \( \frac{dx}{dt} = (x+2)(x-2)^2 \), each critical point is assessed for stability based on how slight deviations impact \( \frac{dx}{dt} \).

**Analyzing \( x = -2 \):**
- The point is semistable because the expression includes \((x-2)^2\), which is always non-negative, and \(x+2 = 0\) at this point.
- Small deviations result in movement back towards or away from \( x = -2 \), but never a consistent tendency, hence semistability.

**Analyzing \( x = 2 \):**
- At this critical point, a slight disturbance leads to \( \frac{dx}{dt} \) becoming positive if \( x > 2 \) and negative if \( x < 2 \).
- This means \( x = 2 \) is an unstable point because any slight change causes the value to consistently move away from \( x = 2 \).

To analyze stability, consider the sign and value of \( \frac{dx}{dt} \) around each critical point, guiding whether points are stable, unstable, or semistable.

A slope field is a visual representation to understand the behavior of a differential equation's solutions. By plotting tiny line segments or slopes at various points within the field, we visualize how solutions might behave.
For \( \frac{dx}{dt} = (x+2)(x-2)^2 \), the slope field helps illustrate:

• How lines approach or diverge at \( x = -2 \) and \( x = 2 \).
• Near \( x = -2 \), the slopes are neutral suggesting semistability.
• Near \( x = 2 \), the lines show divergence indicating instability.

To create a slope field:

• Choose a range of \( x \) values.
• Calculate the slope \( \frac{dx}{dt} \) at each point.
• Plot these as small lines, creating a "field" that reveals overall behavior.

By using slope fields, one can intuitively grasp potential paths of solution curves without solving the differential equation explicitly. This visual tool complements theoretical analysis, offering an immediate understanding of dynamics.