Solve the following differential equation: `(dy)/(dx)=sec(x+y)`

Explanation

Setting x+y-t, we have no differentiation w.r.t. x
`1+(dy)/(dx) = (dt)/(dx)`
Our equation not reads
`(dt)/(dx) - 1 = sec t`
`rArr (dt)/(dx) = 1+ sec t`
`rArr (dt)/(1+ sec t) = dx`
`rArr (cos t dt)/(1+cos t) = dx`
`rArr ((1+ cos t)-1)/(1+ cos t) dt = dx`
`rArr dt - (dt)/(1+ cos t) = dx`
`rArr dt - (dt)/(2 cos^(2).(t)/(2)) = dx`
`rArr dt-(1)/(2)sec^(2).(t)/(2) dt = dx`
Integrating, we get,
`t- tan.(t)/(2) = x + k`
`rArr x+y - tan.(x+y)/(2) =x+k`
`:. y - tan.(x+y)/(2) = k`, k being the constant of integration.