If a curve y=f(x) passes through (1,-1) and satisfies y(1+xy),dx=x,dy, then f(-12) is:

Question

If a curve $y=f(x)$ passes through $(1,-1)$ and satisfies $y(1+xy)\,dx=x\,dy$, then $f\left(-\dfrac12\right)$ is:

Accepted Answer

$$-\frac{x\,dy-y\,dx}{y^2} = x\,dx \implies -\frac{x}{y}=\frac{x^2}{2}+C.$$

Using $(1,-1)$: $C=\dfrac12$. So $y=\dfrac{-2x}{x^2+1}$. At $x=-1/2$: $y=\dfrac{1}{5/4}=\dfrac45$.

Solutions · 1