(i) If a differential equation is ,

then y(I.F) = ∫Q.(I.F)dx + c, where I.F = e^∫Pdx

Given:-

This is a linear differential equation, comparing it with

P = tanx, Q = cosx

I.F = e^∫Pdx

= e^∫tanxdx

= e^log|secx|

= secx

Solution of the equation is given by

y(I.F) = ∫Q.(I.F)dx + c

⇒ ysecx = –∫cosx.secxdx + c

⇒ y = xcosx + Ccosx