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Linear Differential Equations of First Order

If a curve y ...

Question

If a curve y = f(x) passes through the point (1, -1) and satisfies the differential equation, y(1+xy) dx = x dy, then $f (- \frac{1}{2})$ is equal to :

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Solution

The correct option is C

ydx – xdy = $- y^{2}$ xdx

$\Rightarrow \frac{y d x - x d y}{y^{2}} = - x d x \Rightarrow d (\frac{x}{y}) = - x d x$

On integrating both sides

$\frac{x}{y} = \frac{- x^{2}}{2} + c$

it passes through (1, –1)

$\Rightarrow - 1 = \frac{1}{2} + c \Rightarrow x = \frac{- 1}{2} S o, \frac{x}{y} = \frac{- x^{2}}{2} - \frac{1}{2} \Rightarrow y = \frac{- 2 x}{x^{2} + 1} i . e ., f (- \frac{1}{2}) = \frac{4}{5}$

Suggest Corrections

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