The solution of the differential equation ` (dy)/(dx) - (tany)/x = (tan y sin y)/(x^(2))` is

Options

A. `x/(siny) + log x = C`
B. `y/(sin x)+log x =C`
C. `log x +x = C`
D. `log x +y = C`

Answer

b

Explanation

Given , ` (dy)/(dx) - (tany)/x = (tany sin y )/(x^(2))`
` rArr coty " cosec " y (dy)/(dx) - ("cosec "y)/x = 1/(x^(2))`
Put ` - " cosec " y = t`
` rArr cot y " cosec " y (dy)/(dx) = (dt)/(dx)`
Eq. (I ) reduces to , ` (dt)/(dx) +t/x = 1/(x^(2))`
` IF = e^(int P dx) = e^(int 1/x dx)= x`
` :. ` Required solution is given by
` tx = int x * 1/(x^(2)) dx- C`
`rArr - "cosec" y * x = log x = C`
` rArr x/(sin y) + log x = C `