If `xsin((y)/(x))dy=[ysin((y)/(x))-x]dx and y(1)=(pi)/(2)`, then the value of `cos((y)/(x))` is equal to

Options

A. x
B. `(1)/(x)`
C. `logx`
D. `e^(x)`

Answer

C

Explanation

Given,
`(dy)/(dx)=(ysin((y)/(x))-x)/(xsin((y)/(x)))=((y)/(x)sin((y)/(x))-1)/(sin((y)/(x)))`
Put `(y)/(x)=u`
`implies(dy)/(dx)=x(du)/(dx)+u`
`thereforex(du)/(dx)+u=(usinu-1)/(sinu)`
`implies-sinudu=(1)/(x)dx`
`impliescosu=logx+c`
`impliescos((y)/(x))=logx+c`
`becausey(1)=(pi)/(2)`
`therefore"cos"(pi)/(2)=log1+c`
thus, cos `((y)/(x))=logx`