Write the integrating factor of the following differential equation.
$(1 + y^{2}) + (2 x y - cot y) \frac{d y}{d x} = 0$ .

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Solution

divide by $\frac{d y}{d x} (1 + y^{2})$ , we get
$\frac{d y}{d x} + \frac{2 x y - cot y}{1 + y^{2}} = 0 \frac{d x}{d y} + \frac{2 x y}{1 + y^{2}} = \frac{cot y}{1 + y^{2}}$
Here , $P = \frac{2 x y}{1 + y^{2}}$
$I F = e^{\int P d y} I F = e^{\int \frac{2 y}{1 + y^{2}} d y} = e^{log (1 + y^{2})} = 1 + y^{2}$
So, Integrating factor $= 1 + y^{2}$

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