Answer :

Given (x + tan y)dy = sin 2y dx









This is a first order linear differential equation of the form



Here, P = –cosec 2y and


The integrating factor (I.F) of this differential equation is,





We have








[ m log a = log am]



[ elog x = x]


Hence, the solution of the differential equation is,





Let tan y = t


sec2y dy = dt [Differentiating both sides]


By substituting this in the above integral, we get



Recall








[ t = tan y]


Thus, the solution of the given differential equation is


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