Question

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

Answer 1

$(1+y^2)+(2xy-\cot y)\frac{dy}{dx}=0$

On rearranging, the equation becomes:

$\Rightarrow \frac{dx}{dy}+\frac{2y}{1+y^2}x=\frac{\cot y}{1+y^2}$.

This equation is in the form of $\frac{dx}{dy}+P\left(y\right).x=Q\left(y\right)$ where

$P\left(y\right)=\frac{2y}{1+y^2}$

This is a first order linear non-homogeneous differential equation. Thus integrating factor is:

$I.F.=e^{\int P\left(y\right)dy}=e^{\int \frac{2y}{1+y^2}dy}=e^{ln(1+y^2)}=1+y^2$.