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We are given an initial value problem:

\[
\pi y' = -\pi^2 y^2 + 1, \quad y(0) = -0.5
\]

We are tasked with finding the largest interval of \( t \) for which this problem has a unique solution.

### Step-by-step analysis:

1. **Rewrite the equation**: 
   First, simplify the differential equation:
   
   \[
   y' = -\pi y^2 + \frac{1}{\pi}
   \]

   This is a first-order, nonlinear ordinary differential equation (ODE). The existence and uniqueness of the solution for such equations are governed by the **Picard-Lindelöf theorem** (also known as the **Cauchy-Lipschitz theorem**), which states that if the function \( f(t, y) \) and its partial derivative with respect to \( y \) are continuous in some region, then there exists a unique solution in a neighborhood around the initial condition.

2. **Examine the function \( f(y) \)**:
   In this case, the function on the right-hand side of the differential equation is:
   
   \[
   f(y) = -\pi y^2 + \frac{1}{\pi}
   \]

   This is continuous for all \( y \), so we can check its derivative to ensure it satisfies the conditions for uniqueness.

3. **Derivative of \( f(y) \)**:
   Compute the derivative with respect to \( y \):
   
   \[
   \frac{d}{dy}\left( -\pi y^2 + \frac{1}{\pi} \right) = -2\pi y
   \]

   This derivative is also continuous for all values of \( y \). Therefore, the function \( f(y) \) satisfies the conditions for a unique solution.

4. **Initial Condition**:
   The initial condition is given as \( y(0) = -0.5 \).

5. **Potential Singularities**:
   To determine the largest interval, we need to examine potential singularities of the equation. In this case, the singularities occur when \( y \to \infty \) because the equation involves \( y^2 \), but for moderate values of \( y \), no singularities arise.

### Conclusion:
Given that both \( f(y) \) and its derivative are continuous for all real values of \( y \), there are no singularities or discontinuities that would restrict the solution to a specific interval.

Thus, the largest interval of \( t \) for which the initial value problem has a unique solution is:

\[
(-\infty, \infty)
\]

Would you like more details or have any other questions?

### Follow-up Questions:
1. What is the Picard-Lindelöf theorem, and how does it guarantee uniqueness?
2. Can this type of ODE always be solved analytically, or are numerical methods sometimes required?
3. How would you solve this equation using separation of variables?
4. What is the geometric interpretation of this differential equation?
5. How does the initial condition affect the behavior of the solution?

### Tip:
When solving first-order ODEs, always check for the continuity of both the function and its derivative to ensure the existence of a unique solution.

State the largest interval of t for which the given initial value problem has a unique solution: πy' = -π²y² + 1, y(0) = -0.5.

Explore the unique solution interval for the differential equation πy' = -π²y² + 1 with the initial condition y(0) = -0.5. This complex problem involves concepts from differential equations and nonlinear dynamics, suitable for university-level studies.

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