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?

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.