If `y(x)` is the solution of the differential equation `(dy)/(dx)=-2x (y-1)` with `y(0)=1`, then `lim_(x rarr oo) y(x)` equals

Answer

1

Explanation

`(dy)/(dx)=-2x(y-1)`
`(dy)/(dx)+2xy =2x`
I.F. `=e^(int 2xdx)=e^(x^(2))`
solution `implies y.e^(x^(2))= int 2xe^(x^(2)) dx +c`
`y.e^(x^(2))=e^(x^(2))+c` ...(1)
Given `y(0)=1`
`implies 1.1=1+c implies " "c=0`
`:. y. e^(x^(2))=e^(x^(2))` (from (1))
`y=1`
`:. lim_(x rarr oo) y(x)= lim_(x rarr oo) (1)=1`