The solution of the differential equation `dy/dx=sin(x+y)+cos(x+y)` is:

Options

A. ` log [ 1+ tan ((x+y)/2) ] +C = 0`
B. ` log [ 1+ tan ((x+y)/2)] = x+C`
C. ` log [ 1- tan ((x+y)/2)]= x+C`
D. None of the above

Answer

b

Explanation

Put `x+y=v and 1+(dy)/(dx) = (dv)/(dx)`
Therefore, the differential equation reduces to
` (dv)/(dx) = (1+cosv )+sinv`
` = 2 cos^(2). v/2 +2 sin.v/2 cosv/2 = 2 cos^(2).v/2 (1+tan.v/2)`
` int(sec^(2)(v/2)dv)/(2[1+tan(v/2)])=int dx`
` :. log[ 1+tan((x+y)/2)] = x+C`