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

Options

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

Answer

c

Explanation

Given ` (dy)/(dx) = (ysin(y/x)-x)/(x sin(y/x))= (y/x sin(y/x)-1)/(sin(y/x))`
Put ` y/x = u`
` rArr (dy)/(dx) = x(du)/(dx) +u`
` :. x (du)/(dx) + (u (u sin u-1))/(sinu)`
` rArr - sin" u "du = 1/x dx`
On integrating , we get
` rArr cos u = log x + C`
` rArr cos (y/x) = log x+C " " [ :' u = y/x]`
` y (1) = pi/2`
` :. cos.pi/2 = log 1 +C`
` rArr C = 0`
Thus `cos.(y/x)= log x`