If \(y = \left[x + \sqrt{x^2 - 1}\right]^{15} + \left[x - \sqrt{x^2 - 1}\right]^{15}\), then \((x^2 - 1)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}\) is equal to :
Show Hint
- \(125\,y\)
- \(224\,y^2\)
- \(225\,y^2\)
- \(225\,y\)
The Correct Option is D
Solution and Explanation
\(y = u^{15} + v^{15}\), so \(y' = \frac{15u^{15}}{s} - \frac{15v^{15}}{s} = \frac{15(u^{15} - v^{15})}{s}\).
Hence \(s\,y' = 15(u^{15} - v^{15})\). Differentiate: \(s\,y'' + \frac{x}{s}y' = 15\left(\frac{15u^{15}}{s} + \frac{15v^{15}}{s}\right) = \frac{225\,y}{s}\).
Multiply by \(s\): \((x^2 - 1)y'' + x\,y' = 225\,y\).
Answer: (4) \(225\,y\)
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