https://file.aiquickdraw.com/mathbot/file/1726032707490b6aho6nz.jpg

To solve the initial value problem provided in the image, we will analyze the differential equation:

\[
y' = \frac{|y|}{x - 8}, \quad y(x_0) = y_0
\]

### Part (a) A Solution Exists

For a solution to exist, we need to check the conditions where the differential equation is continuous. The right-hand side of the differential equation is \(\frac{|y|}{x - 8}\). This function is continuous as long as \(x \neq 8\). Thus, the initial condition \(x_0\) must be in the domain where the function is defined, meaning \(x_0 \neq 8\).

So, \(x_0\) can be in the interval:

\[
x_0 \in (-\infty, 8) \cup (8, \infty)
\]

For \(y_0\), there are no additional constraints because \(|y|\) is continuous for all \(y\). Hence, \(y_0\) can be any real number:

\[
y_0 \in (-\infty, \infty)
\]

### Part (b) A Solution Exists and is Unique

For uniqueness, we apply the **Picard-Lindelöf theorem** (or existence and uniqueness theorem), which requires that the function and its partial derivative with respect to \(y\) are continuous in some neighborhood of the initial condition \((x_0, y_0)\).

The partial derivative of \(\frac{|y|}{x - 8}\) with respect to \(y\) is:

\[
\frac{\partial}{\partial y} \left(\frac{|y|}{x - 8}\right) = \frac{\text{sgn}(y)}{x - 8}
\]

This derivative is continuous wherever \(x \neq 8\). Hence, for the solution to be unique, \(x_0 \neq 8\) and \(y_0\) can be any real number. Thus, the same intervals apply as in part (a):

\[
x_0 \in (-\infty, 8) \cup (8, \infty), \quad y_0 \in (-\infty, \infty)
\]

### Final Answers:

- **(a) A solution exists provided:**
  \[
  x_0 \in (-\infty, 8) \cup (8, \infty), \quad y_0 \in (-\infty, \infty)
  \]
- **(b) A solution exists and is unique provided:**
  \[
  x_0 \in (-\infty, 8) \cup (8, \infty), \quad y_0 \in (-\infty, \infty)
  \]

Would you like a more detailed explanation of any part of the solution? Here are some related questions:

1. What is the Picard-Lindelöf theorem, and how is it used in determining the uniqueness of solutions?
2. How does the absolute value function affect the continuity of the differential equation?
3. Can you describe a situation where a solution exists but is not unique?
4. What happens if \(x_0 = 8\)? Why can't the solution be determined?
5. How does the sign of \(y_0\) affect the behavior of the solution?

**Tip:** Always ensure to check the continuity and differentiability of the functions involved when dealing with existence and uniqueness theorems.

Consider the initial value problem y' = |y| / (x - 8), y(x0) = y0. Determine the intervals for which (a) a solution exists and (b) a solution exists and is unique.

Explore the conditions for existence and uniqueness of solutions to the initial value problem y' = |y| / (x - 8), with detailed application of the Picard-Lindelöf theorem. Analyze how continuity and differentiability impact the solution.

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