If \( \log_e y = 3 \sin^{-1}x \), then \( (1 - x)^2 y'' - xy' \) at \( x = \frac{1}{2} \) is equal to:
- \( 9e^{\pi/6} \)
- \( 3e^{\pi/6} \)
- \( 3e^{\pi/2} \)
- \( 9e^{\pi/2} \)
The Correct Option is D
Approach Solution - 1
Given:
\[ \ln(y) = 3 \sin^{-1}(x) \]
Differentiating both sides with respect to \(x\):
\[ \frac{1}{y} \cdot y' = 3 \left( \frac{1}{\sqrt{1 - x^2}} \right) \Rightarrow y' = \frac{3y}{\sqrt{1 - x^2}} \]
At \(x = \frac{1}{2}\):
\[ y' = \frac{3e^{3\sin^{-1}\left(\frac{1}{2}\right)}}{\sqrt{1 - \left(\frac{1}{2}\right)^2}} = \frac{3e^{\frac{\pi}{2}}}{\frac{\sqrt{3}}{2}} = 2\sqrt{3} e^{\frac{\pi}{2}} \]
Now, differentiating again to find \(y''\):
\[ y'' = 3 \left( \frac{\sqrt{1 - x^2} \, y' - y \cdot \frac{1}{\sqrt{1 - x^2}}(-2x)}{1 - x^2} \right) \]
Hence,
\[ (1 - x^2)y'' = 3 \left( 3y + \frac{xy}{\sqrt{1 - x^2}} \right) \]
At \(x = \frac{1}{2}\), we have:
\[ y = e^{3\sin^{-1}\left(\frac{1}{2}\right)} = e^{\frac{\pi}{2}} \]
Substitute into the equation:
\[ (1 - x^2)y''\big|_{x=\frac{1}{2}} = 3e^{\frac{\pi}{2}} \left( 3 + \frac{1}{\sqrt{3}} \right) \]
Now compute \((1 - x^2)y'' - xy'\) at \(x = \frac{1}{2}\):
\[ (1 - x^2)y'' - xy' = 3e^{\frac{\pi}{2}}\left( 3 + \frac{1}{\sqrt{3}} \right) - \frac{1}{2}\left( 2\sqrt{3} e^{\frac{\pi}{2}} \right) \]
\[ = e^{\frac{\pi}{2}} \left( 9 + \sqrt{3} - \sqrt{3} \right) = 9e^{\frac{\pi}{2}} \]
Final Answer:
\[ \boxed{9e^{\frac{\pi}{2}}} \]
Approach Solution -2
We are given the equation \( \log_e y = 3 \sin^{-1} x \). We need to find \( (1 - x^2) y'' - xy' \) at \( x = \frac{1}{2} \).
Step 1: Differentiate \( y \)
Start by differentiating \( y = e^{3 \sin^{-1} x} \):
\[ \frac{1}{y} \cdot y' = 3 \cdot \frac{1}{\sqrt{1 - x^2}} \implies y' = \frac{3y}{\sqrt{1 - x^2}} \]At \( x = \frac{1}{2} \), we get:
\[ y' = \frac{3e^{\frac{\pi}{6}}}{\sqrt{3}} = \frac{2\sqrt{3}e^{\frac{\pi}{6}}}{3} \]Step 2: Differentiate \( y' \) to find \( y'' \)
Now differentiate \( y' \) to get \( y'' \):
\[ y'' = \frac{d}{dx} \left( \frac{3y}{\sqrt{1 - x^2}} \right) \]Using the quotient rule, we get:
\[ y'' = 3 \left( \frac{\sqrt{1 - x^2} \cdot y' - y \cdot \left( -\frac{x}{\sqrt{1 - x^2}} \right)}{(1 - x^2)} \right) \]Substitute the expression for \( y' \):
Step 3: Substitute \( x = \frac{1}{2} \)
At \( x = \frac{1}{2} \), we substitute values for \( y \) and \( y' \):
\[ (1 - x^2) y'' = 3 \left( 3e^{\frac{\pi}{6}} + \frac{e^{\frac{\pi}{6}}}{\sqrt{3}} \right) = 3e^{\frac{\pi}{6}} \left( 3 + \frac{1}{\sqrt{3}} \right) \]Now, calculate \( (1 - x^2) y'' - xy' \):
\[ (1 - x^2) y'' - xy' = 3e^{\frac{\pi}{6}} \left( 3 + \frac{1}{\sqrt{3}} \right) - \frac{1}{2} \times \frac{2\sqrt{3}e^{\frac{\pi}{6}}}{3} \]After simplifying:
\[ (1 - x^2) y'' - xy' = 9e^{\frac{\pi}{2}} \]Thus, the correct answer is:
\( \boxed{9e^{\frac{\pi}{2}}} \)
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