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The problem you're referring to involves a first-order differential equation and the application of a uniqueness theorem. The goal is to determine where in the \( t \)-\( y \) plane the hypotheses of the theorem (which guarantee a unique solution) are satisfied.

### Theorem Overview:
- You are given the differential equation \( y' = \frac{t - y}{2t + 7y} \).
- The theorem states that for a unique solution to exist, the function \( f(t, y) = \frac{t - y}{2t + 7y} \) and its partial derivative with respect to \( y \) must be continuous within a certain region.

### Key Steps:
1. **Ensure Continuity of \( f(t, y) \):**
   - The function \( f(t, y) = \frac{t - y}{2t + 7y} \) will be continuous where the denominator \( 2t + 7y \neq 0 \).
   - So, the critical condition for the continuity of \( f(t, y) \) is \( 2t + 7y \neq 0 \).

2. **Partial Derivative of \( f \) with Respect to \( y \):**
   - Next, you need to calculate \( \frac{\partial f}{\partial y} \) and check where it remains continuous.
   - The partial derivative is:
     \[
     \frac{\partial f}{\partial y} = \frac{-1(2t + 7y) - (t - y)(7)}{(2t + 7y)^2}.
     \]
   - This also depends on \( 2t + 7y \neq 0 \), since the denominator must not vanish for continuity.

### Hypothesis for the Theorem:
- The condition \( 2t + 7y \neq 0 \) ensures that both \( f \) and \( \frac{\partial f}{\partial y} \) are continuous.
- Thus, the hypotheses of the theorem are satisfied where \( 2t + 7y \neq 0 \), or equivalently:
  \[
  2t + 7y \neq 0 \quad \text{(which simplifies to)} \quad t \neq -\frac{7}{2}y.
  \]
  
### Conclusion:
- The correct choice corresponds to **\( 2t + 7y \neq 0 \)**.

Do you have any further questions or would you like more details?

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Here are 5 related questions you might find useful:
1. How does the continuity of a function affect the existence of solutions to differential equations?
2. What are other conditions for the uniqueness of solutions in first-order ODEs?
3. How do you interpret the condition \( 2t + 7y \neq 0 \) in terms of geometry in the \( t \)-\( y \) plane?
4. What happens if \( 2t + 7y = 0 \)? What does that imply for the solution behavior?
5. How does one compute partial derivatives of more complex functions involving products and quotients?

**Tip:** When dealing with differential equations, always check both the function and its partial derivatives for continuity to ensure the existence and uniqueness of solutions.

State where in the t-y plane the hypotheses of the theorem are satisfied for the differential equation y' = (t - y) / (2t + 7y).

This problem explores where the hypotheses of the existence and uniqueness theorem for first-order differential equations are satisfied in the t-y plane. It involves analyzing the continuity of the function y' = (t - y) / (2t + 7y) and its partial derivative. Suitable for undergraduate students.

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