Thank you for the clarification. Let's solve the differential equation:

\[
9y'' - 12y' + 4y = 0
\]

### 1. Characteristic Equation
The characteristic equation corresponding to the differential equation is:

\[
9r^2 - 12r + 4 = 0
\]

### 2. Solve the Characteristic Equation
We solve this quadratic equation using the quadratic formula:

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 9 \), \( b = -12 \), and \( c = 4 \). So,

\[
r = \frac{12 \pm \sqrt{(-12)^2 - 4(9)(4)}}{2(9)} = \frac{12 \pm \sqrt{144 - 144}}{18}
\]
\[
r = \frac{12 \pm \sqrt{0}}{18} = \frac{12}{18} = \frac{2}{3}
\]

### 3. General Solution
Since we have a **repeated real root** \( r = \frac{2}{3} \), the general solution is:

\[
y(t) = (C_1 + C_2 t) e^{\frac{2}{3} t}
\]

### 4. Apply Initial Conditions
To determine the constants \( C_1 \) and \( C_2 \), we need the initial conditions provided in the problem. However, these are not specified in the image or the follow-up. If they are:

- \( y(0) = y_0 \)
- \( y'(0) = y'_0 \)

We can substitute these values into the general solution and its derivative to solve for \( C_1 \) and \( C_2 \).

### Example Initial Conditions
Let's assume the initial conditions are:

- \( y(0) = 3 \)
- \( y'(0) = -1 \)

Then,

1. **At \( t = 0 \):**
\[
y(0) = (C_1 + C_2 \cdot 0) e^{0} = C_1 = 3
\]

2. **First derivative \( y'(t) \):**
\[
y'(t) = \frac{d}{dt} \left( (C_1 + C_2 t) e^{\frac{2}{3} t} \right)
\]
Using the product rule:
\[
y'(t) = \left( C_2 \cdot e^{\frac{2}{3} t} \right) + \left( C_1 + C_2 t \right) \cdot \frac{2}{3} e^{\frac{2}{3} t}
\]
\[
y'(t) = e^{\frac{2}{3} t} \left( C_2 + \frac{2}{3} C_1 + \frac{2}{3} C_2 t \right)
\]

At \( t = 0 \):
\[
y'(0) = C_2 + \frac{2}{3} C_1 = -1
\]
Substitute \( C_1 = 3 \):
\[
C_2 + 2 = -1 \implies C_2 = -3
\]

Thus, the specific solution with these initial conditions would be:

\[
y(t) = (3 - 3t) e^{\frac{2}{3} t}
\]

### 5. Analyze Behavior for \( S \)
Given these constants, you can analyze the behavior of the solution \( y(t) \) over time. This would help determine the set \( S \) based on specific criteria (e.g., when \( y(t) \) becomes negative).

---

Do you need further details or additional examples?

### Follow-Up Questions:
1. How does the repeated root affect the form of the general solution compared to distinct roots?
2. What happens to the solution as \( t \) increases for the given example?
3. How would the solution change if the initial conditions were different?
4. What are some methods to verify the correctness of the solution?
5. Can you relate this solution form to physical systems like damped harmonic oscillators?

**Tip:** For repeated real roots in second-order linear differential equations, always remember to include a \( t \) term multiplying one of the exponential solutions to account for the repeated root.

Learn how to solve the differential equation 9y'' - 12y' + 4y = 0 using the characteristic equation and find the general solution. Understand the implications of repeated real roots in second-order linear equations. Suitable for advanced undergraduate students and those interested in differential equations.

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