If a curve `y=f(x)` passes through the point `(1,-1)` and satisfies the differential equation `,y(1+x y)dx""=x""dy` , then `f(-1/2)` is equal to:

Options

A. `-4/5`
B. `2/5`
C. `4/5`
D. `-2/5`

Answer

C

Explanation

`y(1+xy)dx=xdy`
`rArr ydx-xdy+xy^(2)dx=0`
`rArr y^(2)d(x/y)+xy^(2)dx=0`
`rArr x/y+x^(2)/2=C`………….(1)
Curve passes through `(1,-1)`
`rArr -1+1/2=C` (form 1)
`rArr C=-1/2`
`rArr C=-1/2`
`rArr x/y+x^(2)/2=-1/2`
Now put `x=-1/2`
`rArr (-1/2)/(y)+(1/4)1/(2)=-1/2`
`rArr -1/(2y)=-1/2-1/8`
`y=4/5`