The solution of the differential equation `x^2(dy)/(dx)cos(1/x)-ysin(1/x)=-1,` where `y--1` as`x -oo` is

Options

A. `y=sin(1/x)-cos(1/x)`
B. `y=(x+1)/(xsin(1/x))`
C. `y=cos(1/x)+sin(1/x)`
D. `y=(x+1)/(xcos(1//x))`

Answer

A

Explanation

`x^(2)(dy)/(dx)cos(1/x)-ysin(1/x)=-1`
or `(dy)/(dx) -y/x^(2)tan(1/x)=-sec(1/x)1/x^(2)`(linear)
I.F. `=e^((-int(1/x)^(2)tan(1/x)dx))=sec(1/x)`
Thus, solution is y sec `1/x = -intsec^(2)(1/x)1/x^(2)dx=tan(1/x)+c`
Given `yto -1, x to infty`. Thus, `c=-1`
Hence, equation of curve is `y=sin(1/x)-cos(1/x)`