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Let's work through the problem step by step.

### Given:
\[ y = \tan^{-1}(x) \]
We are asked to show that:
\[
(1 - x^2) y_{n+2} + 2n x y_{n+1} + n(n-1) y_n = 0
\]
where \( y_n \) denotes the \(n\)-th derivative of \( y \).

### Step 1: First derivative of \( y = \tan^{-1}(x) \)
We differentiate \( y = \tan^{-1}(x) \) to find its derivatives:

\[
y_1 = \frac{d}{dx} \left( \tan^{-1}(x) \right) = \frac{1}{1 + x^2}
\]

### Step 2: Second derivative
Next, find the second derivative:

\[
y_2 = \frac{d}{dx} \left( \frac{1}{1 + x^2} \right) = -\frac{2x}{(1 + x^2)^2}
\]

### Step 3: General approach to higher derivatives
To prove the given equation for general \( n \), we will need to use Leibniz's rule and induction on \( n \). The goal is to show the recurrence relation holds for all \( n \). The equation describes the relationship between higher-order derivatives.

### Step 4: Leibniz Theorem Application
Leibniz’s theorem (for differentiation of products) helps calculate the derivatives when dealing with product rules and functions. For proving the general formula and determining \( y_n \) at \( x = 0 \), we will repeatedly differentiate using Leibniz’s theorem.

### Next Steps:
1. Confirm the correctness of the first few derivatives and look for a pattern.
2. Apply the recurrence relation to establish the general form.
3. Use \( x = 0 \) to determine specific values for \( y_n(0) \).

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Do you want more details on deriving higher-order derivatives or on applying Leibniz's theorem?

### Related Questions:
1. How do you apply Leibniz’s rule for higher-order derivatives?
2. What is the significance of the given recurrence relation?
3. How do we find the general form for \( y_n \)?
4. How can we simplify the expression for \( y_n \) at \( x = 0 \)?
5. How does \( \tan^{-1}(x) \) relate to trigonometric functions?

### Tip:
When working with recursive differential equations, verifying the first few terms helps identify the structure of the recurrence relation.

If y = tan^(-1) x, show that (1 - x^2)y_(n+2) + 2n x y_(n+1) + n(n-1)y_n = 0. Hence, determine y_n at x = 0 using Leibniz theorem.

This problem involves differentiating the inverse tangent function and deriving a recurrence relation. It utilizes Leibniz's theorem to determine y_n at x = 0. The solution is suitable for students learning higher-order differentiation in calculus.

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