The equation from the image is as follows:

$(1 - x^2)y_{n+2} - (2n+1)xy_{n+1} - n^2 y_n = 0$ where $y = (\sin^{-1} x)^2$.

The task is to:

1. Find the recurrence relation for $y_n$.
2. Find $y_n$ when $n = 0$.

Let me walk through the process:

Step 1: Analyzing the recurrence relation

We are provided with the recurrence relation in terms of $y_n$, $y_{n+1}$, and $y_{n+2}$. To solve this, we will need to determine values for each of these components or work with the functional form of $y = (\sin^{-1} x)^2$.

Step 2: Finding $y_n$

To proceed with calculating specific values like $y_0$, we might need to assume initial conditions for $y_n$ or find a general form. I’ll now calculate $y_0$.

Let me calculate that for you.The value of $y_0$, given that $y = (\sin^{-1} x)^2$, is 0.

Would you like further details or explanations on any part of the process?

Here are 5 related questions to expand on the topic:

1. How do we solve for $y_n$ for other values of $n$ using the recurrence relation?
2. What is the importance of initial conditions in solving recurrence relations?
3. Can we derive a general solution for $y_n$ from the given recurrence relation?
4. How does the function $(\sin^{-1} x)^2$ behave for different ranges of $x$?
5. What are the applications of solving recurrence relations in mathematical physics?

Tip: When solving recurrence relations, understanding the initial conditions and patterns in the relation often leads to identifying the closed-form solution.