The solution of the differential equation $y d x - (x + 2 y^{2}) d y = 0$ is $x = f (y)$ . If $f (- 1) = 1$ , then $f (1)$ is equal to:

2

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3

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4

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1

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Solution

The correct option is A $3$
Given $y d x - (x + 2 y^{2}) d y = 0$
$\Rightarrow y d x - x d y = 2 y^{2} d y$
$\Rightarrow \frac{y d x - x d y}{y^{2}} = 2 d y$
$\Rightarrow d (\frac{x}{y}) = 2 d y$
Integrating we get,
$\Rightarrow (\frac{x}{y}) = 2 y + c \Rightarrow 2 y^{2} + c y = x = f (y)$
Given $f (- 1) = 1 \Rightarrow 2 (- 1)^{2} - c = 1 \Rightarrow c = 1$
$\Rightarrow f (y) = 2 y^{2} + y$
$∴ f (1) = 2 (1)^{2} + 1 = 3$

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