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para esta ecuacion cual es el intervalo apropiado

The given differential equation is:

\[
\frac{dy}{dt} + 20y = 24
\]

with the particular solution:

\[
y = \frac{6}{5} - \frac{6}{5} e^{-20t}
\]

To determine the appropriate interval for \( t \), we analyze the solution provided. 

### Analysis of the Solution:

1. **General Behavior**: The term \( e^{-20t} \) suggests that the function includes an exponential decay component. As \( t \) increases, \( e^{-20t} \) approaches zero.

2. **Initial Condition**: Since the solution is given explicitly, it suggests the interval starts from \( t = 0 \) (assuming this is the initial condition), as \( t < 0 \) could imply non-physical (negative) time for many real-world applications.

3. **Long-term Behavior**: As \( t \to \infty \), \( e^{-20t} \to 0 \), and the solution approaches \( y = \frac{6}{5} \). 

### Conclusion:

The natural interval for this solution is \( t \geq 0 \) because the exponential term ensures the solution remains finite and well-behaved for all non-negative \( t \). 

Therefore, the appropriate interval for \( t \) is:

\[
t \in [0, \infty)
\]

This interval ensures that \( y(t) \) remains real, non-negative, and bounded, consistent with the nature of the differential equation and its solution.

---

Would you like further details or have any questions?

### Related Questions:

1. How do we find the general solution for the given differential equation?
2. What are the initial conditions implied by the particular solution provided?
3. How does the behavior of the solution change as \( t \) approaches infinity?
4. Can the interval \( t \) be negative, and what would be the implications?
5. How can we verify that \( y = \frac{6}{5} - \frac{6}{5} e^{-20t} \) is indeed a solution to the differential equation?

### Tip:

Always check the behavior of exponential functions in differential equations to understand the long-term behavior of the solution.

Understand how to determine the appropriate interval for the solution of a differential equation involving exponential decay. Learn about differential equation forms, exponential functions, and their behavior over time. Suitable for advanced mathematics students and enthusiasts.

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