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Class-12-science»Maths

if y=sqrt(x+1)-sqrt(x-1), prove that (x²-1)d²y/dx²+x dy/dx - y/4=0

y = \sqrt{x + 1} - \sqrt{x - 1} . . . . . . . (2) S o \frac{d y}{d x} = \frac{1}{2 \sqrt{x + 1}} - \frac{1}{2 \sqrt{x - 1}} . . . . . . (1) = \frac{\sqrt{x - 1} - \sqrt{x + 1}}{4 \sqrt{x^{2} - 1}} \frac{d y}{d x} = \frac{\sqrt{x - 1} - \sqrt{x + 1}}{4 \sqrt{x^{2} - 1}} \frac{d^{2} y}{d x^{2}} = \frac{\sqrt{x^{2} - 1} (\frac{1}{2 \sqrt{x - 1}} - \frac{1}{2 \sqrt{x + 1}}) - (\sqrt{x - 1} - \sqrt{x + 1}) \times \frac{x}{\sqrt{x^{2} - 1}}}{4 (x^{2} - 1)} 4 (x^{2} - 1) \frac{d^{2} y}{d x^{2}} = \sqrt{x - 1} \times \sqrt{x + 1} (\frac{1}{2 \sqrt{x - 1}} - \frac{1}{2 \sqrt{x + 1}}) - (\sqrt{x - 1} - \sqrt{x + 1}) \times \frac{x}{\sqrt{x - 1} \times \sqrt{x + 1}} 4 (x^{2} - 1) \frac{d^{2} y}{d x^{2}} = \frac{1}{2} [\sqrt{x + 1} - \sqrt{x - 1}] - x (\frac{1}{\sqrt{x + 1}} - \frac{1}{\sqrt{x - 1}}) 4 (x^{2} - 1) \frac{d^{2} y}{d x^{2}} = \frac{y}{2} - 2 x \frac{d y}{d x} (x^{2} - 1) \frac{d^{2} y}{d x^{2}} + \frac{x}{2} \frac{d y}{d x} - \frac{y}{8} = 0

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