The differential equation having `y=(sin^(-1)x)^(2)+A(cos^(-1)x)+B`, where A and B are abitary constant , is

Options

A. `(1-x^(2))y_(2)-xy_(1)=2`
B. `(1-x^(2))y_(2)+yy_(1)=0`
C. `(1-x^(2))y_(2)+xy_(1)=0`
D. None of these

Answer

To find the differential equation for the function \( y = (\sin^{-1} x)^2 + A(\cos^{-1} x) + B \), where \( A \) and \( B \) are arbitrary constants, we will follow these steps:

### Step 1: Differentiate \( y \) with respect to \( x \)

We start with the given function:
\[ y = (\sin^{-1} x)^2 + A(\cos^{-1} x) + B \]

Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 2(\sin^{-1} x) \cdot \frac{d}{dx}(\sin^{-1} x) + A \cdot \frac{d}{dx}(\cos^{-1} x) \]

Using the derivatives:
\[ \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} \quad \text{and} \quad \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}} \]

We substitute these into our equation:
\[ \frac{dy}{dx} = 2(\sin^{-1} x) \cdot \frac{1}{\sqrt{1 - x^2}} - \frac{A}{\sqrt{1 - x^2}} \]
\[ \frac{dy}{dx} = \frac{2(\sin^{-1} x) - A}{\sqrt{1 - x^2}} \]

### Step 2: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

Now we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{2(\sin^{-1} x) - A}{\sqrt{1 - x^2}} \right) \]

Using the quotient rule:
\[ \frac{d^2y}{dx^2} = \frac{\left(2 \cdot \frac{d}{dx}(\sin^{-1} x) \cdot \sqrt{1 - x^2} - (2(\sin^{-1} x) - A) \cdot \frac{d}{dx}(\sqrt{1 - x^2})\right)}{(1 - x^2)} \]

Calculating \( \frac{d}{dx}(\sqrt{1 - x^2}) = -\frac{x}{\sqrt{1 - x^2}} \):
\[ \frac{d^2y}{dx^2} = \frac{2 \cdot \frac{1}{\sqrt{1 - x^2}} \cdot \sqrt{1 - x^2} - (2(\sin^{-1} x) - A) \cdot \left(-\frac{x}{\sqrt{1 - x^2}}\right)}{1 - x^2} \]
\[ = \frac{2 - \frac{x(2(\sin^{-1} x) - A)}{\sqrt{1 - x^2}}}{1 - x^2} \]

### Step 3: Form the differential equation

Now we can express the second derivative in terms of the first derivative and \( y \):
Let \( y_1 = \frac{dy}{dx} \) and \( y_2 = \frac{d^2y}{dx^2} \).

We can rearrange the terms to form the differential equation:
\[ (1 - x^2)y_2 - xy_1 = 2 \]

This gives us the final form of the differential equation:
\[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 2 \]

### Final Answer:
The required differential equation is:
\[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 2 \]