If a curve $y = f(x)$ passes through the point $(1, –1)$ and satisfies the differential equation, $y(1 + xy) dx = x dy$, then $f(-1/2)$ is equal to:
$$ \begin{align} (1) & &4/5\\ (2) & &-2/5\\ (3) & &-4/5\\ (4) & &2/5 \end{align} $$
$(1)$, $(2)$, $(3)$ and $(4)$ are the options. This is a multiple choice question.
My working:
$$ -d(x/y) = xdx $$
Till here I have found out
$y=ux$ and $y(1 + xy) dx = x dy \Rightarrow uxdx+ux^3dx=uxdx+x^2du$
$uxdx+u^2x^3dx=uxdx+x^2du \Rightarrow u^2x^3dx=x^2du$
$u^2x^3dx=x^2du \Rightarrow x dx=\frac{du}{u^2}$
$x dx=\frac{du}{u^2} \Rightarrow \frac{x^2}{2}+c=\frac{-1}{u}$
$\frac{x^2}{2}+c=\frac{-1}{u} \Rightarrow y=\frac{-2x}{x^2+2c}$ and according to "$f(x)$ passes through the point $(1,-1)$"
$c=\frac{1}{2}$ so
$y=\frac{-2x}{x^2+1}$
so $f(-1/2)=4/5$