on which interval is the IVP (sin(x)y'+(xy)/(x-5)=x+1, y(4)=1, guaranteed to have a unique solution?



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.

On which interval is the IVP (sin(x)y' + (xy)/(x-5) = x + 1, y(4) = 1) guaranteed to have a unique solution?

Explore the interval where the initial value problem (IVP) sin(x)y' + (xy)/(x-5) = x + 1 with y(4) = 1 has a unique solution. The analysis includes continuity conditions and the application of the Picard-Lindelöf theorem. Suitable for students in Grades 11-12.

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