If `y=e^(m cos^(-1)x)`, then `(1-x^(2))y_(2)-xy_(1)-m^(2)y` is equal to

Options

A. 0
B. 1
C. `-1`
D. `2m^(2)`

Answer

To solve the problem, we need to find the expression \( (1 - x^2)y_{2} - xy_{1} - m^{2}y \) given that \( y = e^{m \cos^{-1}(x)} \).

### Step 1: Find \( y_1 \) (First Derivative)
We start with:
\[ y = e^{m \cos^{-1}(x)} \]
To find the first derivative \( y_1 = \frac{dy}{dx} \), we apply the chain rule:
\[ y_1 = \frac{d}{dx} \left( e^{m \cos^{-1}(x)} \right) = e^{m \cos^{-1}(x)} \cdot \frac{d}{dx}(m \cos^{-1}(x)) \]
The derivative of \( \cos^{-1}(x) \) is:
\[ \frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1 - x^2}} \]
Thus,
\[ y_1 = e^{m \cos^{-1}(x)} \cdot \left(-\frac{m}{\sqrt{1 - x^2}}\right) = -\frac{m e^{m \cos^{-1}(x)}}{\sqrt{1 - x^2}} \]

### Step 2: Find \( y_2 \) (Second Derivative)
Next, we find the second derivative \( y_2 = \frac{d^2y}{dx^2} \). We differentiate \( y_1 \):
\[ y_2 = \frac{d}{dx}\left(-\frac{m e^{m \cos^{-1}(x)}}{\sqrt{1 - x^2}}\right) \]
Using the quotient rule:
\[ y_2 = \frac{(-m \cdot \frac{d}{dx}(e^{m \cos^{-1}(x)})) \cdot \sqrt{1 - x^2} - (-m e^{m \cos^{-1}(x)}) \cdot \frac{d}{dx}(\sqrt{1 - x^2})}{(1 - x^2)} \]
Calculating \( \frac{d}{dx}(e^{m \cos^{-1}(x)}) \) again gives us:
\[ \frac{d}{dx}(e^{m \cos^{-1}(x)}) = -\frac{m e^{m \cos^{-1}(x)}}{\sqrt{1 - x^2}} \]
And for \( \frac{d}{dx}(\sqrt{1 - x^2}) \):
\[ \frac{d}{dx}(\sqrt{1 - x^2}) = -\frac{x}{\sqrt{1 - x^2}} \]
Substituting these into the expression for \( y_2 \):
\[ y_2 = \frac{(-m)(-\frac{m e^{m \cos^{-1}(x)}}{\sqrt{1 - x^2}}) \cdot \sqrt{1 - x^2} - (-m e^{m \cos^{-1}(x)}) \cdot \left(-\frac{x}{\sqrt{1 - x^2}}\right)}{(1 - x^2)} \]
This simplifies to:
\[ y_2 = \frac{m^2 e^{m \cos^{-1}(x)} - mx e^{m \cos^{-1}(x)}}{(1 - x^2)} \]

### Step 3: Substitute into the Expression
Now we substitute \( y_1 \) and \( y_2 \) into the expression:
\[ (1 - x^2)y_{2} - xy_{1} - m^{2}y \]
Substituting \( y_2 \):
\[ (1 - x^2) \left(\frac{m^2 e^{m \cos^{-1}(x)} - mx e^{m \cos^{-1}(x)}}{(1 - x^2)}\right) - x\left(-\frac{m e^{m \cos^{-1}(x)}}{\sqrt{1 - x^2}}\right) - m^2 e^{m \cos^{-1}(x)} \]
This simplifies to:
\[ m^2 e^{m \cos^{-1}(x)} - mx e^{m \cos^{-1}(x)} + \frac{mx e^{m \cos^{-1}(x)}}{\sqrt{1 - x^2}} - m^2 e^{m \cos^{-1}(x)} \]
The \( m^2 e^{m \cos^{-1}(x)} \) terms cancel out, leading to:
\[ -mx e^{m \cos^{-1}(x)} + \frac{mx e^{m \cos^{-1}(x)}}{\sqrt{1 - x^2}} = 0 \]
Thus, the entire expression equals zero:
\[ (1 - x^2)y_{2} - xy_{1} - m^{2}y = 0 \]

### Final Answer
\[ (1 - x^2)y_{2} - xy_{1} - m^{2}y = 0 \]