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Higher Order Derivatives

If y=xe1/x,...

Question

If $y = x e^{\frac{1}{x}}$ , then $x^{3} y_{2} + x y_{1} =$

A

0

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B

y

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C

- y

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D

\frac{1}{y}

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Solution

The correct option is A $y$
$y_{1} = e^{\frac{1}{x}} - \frac{e^{\frac{1}{x}}}{x}$
$x y_{1} = y - e^{\frac{1}{x}}$
$x y_{2} + y_{1} = y_{1} + \frac{e^{\frac{1}{x}}}{x^{2}}$
$x^{3} y_{2} = e^{\frac{1}{x}}$
$x^{3} y_{2} = y - x y_{1}$
$x^{3} y_{2} + x y_{1} = y$

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