To determine the interval on which the initial value problem (IVP)

$\sin(x) y' + \frac{xy}{x-5} = x + 1, \quad y(4) = 1$

is guaranteed to have a unique solution, we can use the Picard-Lindelöf theorem, which states that if the function and its partial derivative with respect to $y$ are continuous in a region containing the initial point, then a unique solution exists.

Step 1: Rewrite the Equation

First, we rewrite the equation in standard form:

$y' = \frac{(x + 1)(x - 5) - xy \sin(x)}{\sin(x)(x - 5)}$

This shows that $y'$ is a function of $x$ and $y$.

Step 2: Identify Continuity Conditions

1. For the function $\sin(x)$: The function $\sin(x)$ is continuous everywhere.
2. For the term $\frac{xy}{x-5}$: This term has a singularity at $x = 5$, meaning it is not continuous at this point.

Step 3: Determine the Interval

Given the initial condition $y(4) = 1$, the point $x = 4$ is in the interval $(-\infty, 5)$ since $x = 5$ is where the function becomes undefined.

Conclusion

The IVP is guaranteed to have a unique solution in the interval:

$(4, 5)$

Summary

• The interval of uniqueness for the IVP is $(4, 5)$.
• The continuity of the function and its partial derivative is preserved within this interval.

Would you like more details or have any questions? Here are some related questions:

1. What methods can be used to solve the differential equation?
2. How does the Picard-Lindelöf theorem apply to other types of equations?
3. What happens to the solution if we extend the interval beyond $(4, 5)$?
4. Can the initial condition change the interval of uniqueness?
5. How can we analyze the behavior of the solution near the singularity at $x = 5$?

Tip: When analyzing differential equations, always check for points of discontinuity, as they will affect the existence and uniqueness of solutions.