Suppose a solutions of the differential equation `(xy^3 + x^2y^7) dy/dx =1 ` satisfies the initial conditions `y(1/4)=1.` Then the value of `dy/dx` when `y = -1` is

Options

A. `-(3)/(20)`
B. `-(20)/(3)`
C. `-(5)/(16)`
D. `-(16)/(5)`

Answer

D

Explanation

`(dx)/(dy)=xy^(3)+x^(2)y^(7)`
`rArr" "(1)/(x^(2))(dx)/(dy)-(1)/(x)y^(3)=y^(7)`
`rArr" "(d(-1//x))/(dy)-(1)/(x)(y^(3))=y^(7)`
`I.F.=e^(y^(4)//4)`
Solution is `(-e^(y^(4)//4))/(x)=y^(4)e^(y^(4)//4)-4e^(y^(4)//4)-e^(1//4)`
when `y = -1, x = (1)/(4)`
`therefore" "(dy)/(dx) = -(16)/(5)`