(y - sin2x)dx + tanx dy = 0
\(\frac{dy}{dx} = \frac{sin^2x - y}{tanx}\)
⇒ \(\frac{dy}{dx} + \frac y{tanx} = \frac{sin^2x}{tanx} = sinx. cos x\)
which is linear differential equation
\(\therefore I.F = e^{\int P dx}\)
\(= e^{\int\frac1{tanx}dx}\)
\(= e^{\int cot x\,dx}\)
\(= e^{log\, sinx}\)
\(= sin x\)
\(\therefore y \times(I.F) = \int Q \times (I.F)dx\)
\(y \times sin x= \int (sinx \,cos x)sinx\,dx\)
\(= \int sin^2x \,cosx\,dx\)
\(= \frac{sin^3x}{3} + C\)
⇒ \(y = \frac{sin^2x}{3} + \frac C{sin\,x}\)
⇒ \(y = \frac{sin^2x}{3 } + C \,cosecx\)
is solution of given differential equation.
