If a curve y=f(x) passes through the point (1,-1) and satisfies the differential equation,…

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 $$f\left( { - {1 \over 2}} \right)$$ is equal to :

Accepted Answer

$$y\left( {1 + xy} \right)dx = xdy$$

$${{xdy - ydx} \over {{y^2}}} = xdx \Rightarrow \int { - d\left( {{x \over y}} \right) = \int {xdx} } $$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ - {x \over y} = {{{x^2}} \over 2} + C\,\,$$

as $$\,\,\,y\left( 1 \right) =  - 1 \Rightarrow C = {1 \over 2}$$

Hence, $$y = {{ - 2x} \over {{x^2} + 1}} \Rightarrow f\left( {{{ - 1} \over 2}} \right) = {4 \over 5}$$

Solutions · 1