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Logarithmic Differentiation

if, Y = eta...

Question

if, $Y = e^{tan - 1 x}$ prove that $(1 + x^{2}) d^{2} y / d x^{2} + (2 x - 1) d y / d x = 0$

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Solution

$y = e^{t a n - 1} x$
$\frac{d y}{d x} = \frac{e^{t a n - 1} x}{1 + x^{2}}$
$\frac{d^{2} y}{d x^{2}} = \frac{(1 + x^{2}) \frac{e^{t a n - 1} x}{{(1 + x)}^{2}} - e^{t a n - 1} (2 x)}{{(1 + x)}^{2}} = \frac{e^{t a n - 1} (1 - 2 x)}{{(1 + x)}^{2}}$
$= \frac{(1 + x^{2}) \frac{d y}{d x} (1 - 2 x)}{{(1 + x^{2})}^{2}}$
$= \frac{d y}{d x} \frac{(1 - 2 x)}{(1 + x^{2})}$
$\Rightarrow (1 + x^{2}) \frac{d^{2} y}{d x^{2}} + (2 x - 1) \frac{d y}{d x} = 0$

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