\begin{tabular}{l} Given the differential equation \ $$ x^{\prime}=(x+3)(x+1)^{3} x^{2}(x-1.5) $$. \ List the constant (i.e. equilibrium) solutions to \ this differential equation in increasing order and \ indicate whether or not these solutions are \ stable, semi-stable, or unstable. Confirm your \ answer by plotting the slope field. \ \hline stable \end{tabular}

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Alright, I need to analyze the given differential equation $$ x' = (x + 3)(x + 1)^3 x^2 (x - 1.5) $$ to find its constant (equilibrium) solutions, list them in increasing order, and determine their stability. Additionally, I need to confirm the stability by plotting the slope field.

First, let's identify the equilibrium solutions. These are the values of $$ x $$ where $$ x' = 0 $$. So, I'll set the right-hand side of the equation to zero:

$$
(x + 3)(x + 1)^3 x^2 (x - 1.5) = 0
$$

This equation will be zero when any of the factors is zero. Therefore, the equilibrium solutions are:

1. $$ x = -3 $$
2. $$ x = -1 $$ (with multiplicity 3)
3. $$ x = 0 $$ (with multiplicity 2)
4. $$ x = 1.5 $$

Now, I'll list these solutions in increasing order:

1. $$ x = -3 $$
2. $$ x = -1 $$
3. $$ x = 0 $$
4. $$ x = 1.5 $$

Next, I need to determine the stability of each equilibrium solution. To do this, I'll analyze the sign of $$ x' $$ around each equilibrium point.

1. **For $$ x = -3 $$:**
   - To the left of $$ x = -3 $$, all factors $$ (x + 3) $$, $$ (x + 1)^3 $$, $$ x^2 $$, and $$ (x - 1.5) $$ are negative except $$ x^2 $$, which is positive.
   - To the right of $$ x = -3 $$, $$ (x + 3) $$ becomes positive, while the other factors remain the same.
   - The sign of $$ x' $$ changes from negative to positive as $$ x $$ increases through $$ -3 $$, indicating that $$ x = -3 $$ is **unstable**.

2. **For $$ x = -1 $$:**
   - Since $$ (x + 1)^3 $$ has a multiplicity of 3, the behavior around $$ x = -1 $$ is more complex.
   - To the left of $$ x = -1 $$, $$ (x + 1)^3 $$ is negative, and to the right, it becomes positive.
   - The other factors $$ (x + 3) $$, $$ x^2 $$, and $$ (x - 1.5) $$ are positive on both sides.
   - The sign of $$ x' $$ changes from negative to positive as $$ x $$ increases through $$ -1 $$, suggesting that $$ x = -1 $$ is **unstable**.

3. **For $$ x = 0 $$:**
   - $$ x^2 $$ is always positive.
   - $$ (x + 3) $$ is positive to the right of $$ x = -3 $$ and negative to the left.
   - $$ (x + 1)^3 $$ is positive to the right of $$ x = -1 $$ and negative to the left.
   - $$ (x - 1.5) $$ is negative to the left of $$ x = 1.5 $$ and positive to the right.
   - The sign of $$ x' $$ changes from negative to positive as $$ x $$ increases through $$ 0 $$, indicating that $$ x = 0 $$ is **unstable**.

4. **For $$ x = 1.5 $$:**
   - $$ (x - 1.5) $$ is positive to the right of $$ x = 1.5 $$ and negative to the left.
   - The other factors $$ (x + 3) $$, $$ (x + 1)^3 $$, and $$ x^2 $$ are positive on both sides.
   - The sign of $$ x' $$ changes from positive to negative as $$ x $$ increases through $$ 1.5 $$, suggesting that $$ x = 1.5 $$ is **stable**.

To confirm these findings, I'll plot the slope field of the differential equation. The slope field will show the direction of the solutions at various points, which should align with the stability analysis:

- Around $$ x = -3 $$, $$ x = -1 $$, and $$ x = 0 $$, the slopes change from negative to positive, indicating instability.
- Around $$ x = 1.5 $$, the slopes change from positive to negative, indicating stability.

This graphical confirmation supports the analytical determination of the stability of each equilibrium solution.

The equilibrium solutions are: 1. $$ x = -3 $$ (unstable) 2. $$ x = -1 $$ (unstable) 3. $$ x = 0 $$ (unstable) 4. $$ x = 1.5 $$ (stable) **Slope Field Confirmation:** The slope field shows that solutions near $$ x = 1.5 $$ tend to move towards it, confirming its stability, while solutions near the other equilibrium points move away, confirming their instability.

Given the differential equation

.

List the constant (i.e. equilibrium) solutions to

this differential equation in increasing order and

indicate whether or not these solutions are

stable, semi-stable, or unstable. Confirm your

answer by plotting the slope field.

stable

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Given the differential equation . List the constant (i.e. equilibrium) solutions to this differential equation in increasing order and indicate whether or not these solutions are stable, semi-stable, or unstable. Confirm your answer by plotting the slope field. stable