If y = `(cot^(-1) x)^(2) ` then

Options

A. `(x^(2) +1)^(2) (d^(2) y)/(dx^(2)) +x (x^(2) +1) (dy)/(dx) -2 =0`
B. `(x^(2) +1)^(2) (d^(2)y)/(dx^(2)) - 2x (x^(2) +1) (dy)/(dx) +1 =0`
C. `(x^(2) +1)^(2) (d^(2)y)/(dx^(2)) + 2 x(x^(2) +1) (dy)/(dx) -2 =0`
D. `(x^(2)+1)^(2) (d^(2)y)/(dx^(2)) -x (x^(2)+1) +2 =0`

Answer

To solve the problem where \( y = (\cot^{-1} x)^2 \), we need to find the second derivative \( \frac{d^2y}{dx^2} \) and check which of the given options is satisfied by this expression.

### Step-by-Step Solution:

1. **Differentiate \( y \) with respect to \( x \)**:
\[ y = (\cot^{-1} x)^2 \]
Using the chain rule:
\[ \frac{dy}{dx} = 2(\cot^{-1} x) \cdot \frac{d}{dx}(\cot^{-1} x) \]
The derivative of \( \cot^{-1} x \) is:
\[ \frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2} \]
Therefore:
\[ \frac{dy}{dx} = 2(\cot^{-1} x) \cdot \left(-\frac{1}{1 + x^2}\right) = -\frac{2 \cot^{-1} x}{1 + x^2} \]

2. **Square \( \frac{dy}{dx} \)**:
\[ \left(\frac{dy}{dx}\right)^2 = \left(-\frac{2 \cot^{-1} x}{1 + x^2}\right)^2 = \frac{4 (\cot^{-1} x)^2}{(1 + x^2)^2} \]

3. **Multiply by \( (1 + x^2)^2 \)**:
\[ (1 + x^2)^2 \cdot \left(\frac{dy}{dx}\right)^2 = 4 (\cot^{-1} x)^2 \]
Substituting \( y \) for \( (\cot^{-1} x)^2 \):
\[ (1 + x^2)^2 \cdot \left(\frac{dy}{dx}\right)^2 = 4y \]

4. **Differentiate the equation again**:
Using the product rule on the left-hand side:
\[ \frac{d}{dx}\left((1 + x^2)^2 \cdot \left(\frac{dy}{dx}\right)^2\right) = (1 + x^2)^2 \cdot \frac{d}{dx}\left(\left(\frac{dy}{dx}\right)^2\right) + \left(\frac{dy}{dx}\right)^2 \cdot \frac{d}{dx}\left((1 + x^2)^2\right) \]

The derivative of \( (1 + x^2)^2 \) is:
\[ 2(1 + x^2)(2x) = 4x(1 + x^2) \]

The derivative of \( \left(\frac{dy}{dx}\right)^2 \) is:
\[ 2\frac{dy}{dx}\frac{d^2y}{dx^2} \]

So we have:
\[ (1 + x^2)^2 \cdot 2\frac{dy}{dx}\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 \cdot 4x(1 + x^2) = 0 \]

5. **Rearranging the equation**:
We can factor out \( 2\frac{dy}{dx} \) (assuming \( \frac{dy}{dx} \neq 0 \)):
\[ (1 + x^2)^2 \cdot \frac{d^2y}{dx^2} + 2x(1 + x^2)\cdot \frac{dy}{dx} - 2 = 0 \]

### Final Form:
Thus, we arrive at the equation:
\[ (1 + x^2)^2 \cdot \frac{d^2y}{dx^2} + 2x(1 + x^2)\cdot \frac{dy}{dx} - 2 = 0 \]

### Conclusion:
The correct option is **Option 3**.