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Question
Solve `sin x dy/dx - y = sin x.tan x/2`
Solution
`dy/dx - y.cosecx = tan x/2`
`dy/dx - y.cosecx = tan x/2`
`dy/dx - y.cosec x = tan x/2` ....(i)
Compare `dy/dx + Py = Q`
P = -cosecx, Q = `tan x/2`
`I.F. = e^(intPdx)`
`I.F. = e^(int-cosecx dx)`
`I.F. = e^(-log_e(cosecx- cotx))`
`I.F. = e^(log_e(cosecx - cot x)^(-1))`
`I.F. = (cosec x - cot x)^(-1) = (cosect x + cot x)`
∴ solution of the linear differential equation
`y.I.F. = int Q.IF.dx`
`y.(cosecx + cotx) = int tan x/2 ((1+cosx)/sinx) dx`
`y.(cosecx - cot x) = int (sin x/2)/(cos x/2) . (2cos^2 x/2)/(2sin x/2 .cos x/2) dx = int 1 dx`
y.(cosec x + cot x) = x + c
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