Question

solve this problem by the method of separation of variable!!!!!!

Answer 1

$\frac{dy}{dx} = cos(x + y) \\ put \: x + y = u \\ diff \: on \: both \: side \: wrt \: \\ 1 + \frac{dy}{dx} = \frac{du}{dx}$

$\frac{dy}{dx} = 1 - \frac{du}{dx} \\ put \: in \: 1 \\ 1 - \frac{du}{dx} = cos \: \: u$

$\frac{du}{dx} = 1 - cos \: u \\ dx \: = \: \frac{du}{1 - cos \: u} \\ rationlise \: and \: then \: seperate \\ integrtion \: on \: both \: side \\ x = - cot \: u - cosec \: u \: + c \\ put \: u = x + y \\ x = - cot \: (x + y) \: - cosec(x + y) \: + c$