Question

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Answer 1

Step by step explanation:-

$y(1 + xy)dx = xdy$

$\frac{y}{x}(1 + xy) = \frac{dy}{dx}$

$y = vx⇒ \frac{dy}{dx} = v + x \frac{dv}{dx}$

$∴v \: (1 + {vx}^{2} ) = v + x \frac{dv}{dx}$

${v}^{2} {x}^{2} = x \frac{dv}{dx}$

${v}^{2} x = \frac{dv}{dx}$

$f \: xdx = f \frac{1}{ {v}^{2} } dv$

$\frac{ {x}^{2} }{2} = \frac{ - 1}{v} + c$

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

$put \: ( 1, - 1)$

$\frac{ {1}^{2} }{2 } = \frac{ - x}{y} - \frac{1}{2}$

$we \: have \: to \: find \: f (-\frac{1}{2} )$

$substitude \: x = -\frac{1}{2}$

$\frac{ {(-\frac{1}{2})}^{2} }{2} = \frac{ - ( - \frac{1}{2} )}{y} - \frac{1}{2}$

$\frac{1}{8} = \frac{1}{2y} - \frac{1}{2}$

$y = \frac{4}{5}$