Example 3
\[ y^{\prime }+2y\cot \left ( 2x\right ) =4x\csc \left ( x\right ) \sec ^{2}\left ( x\right ) \]
In normal form the ode is
\[ y^{\prime }+p\left ( x\right ) y=q\left ( x\right ) \]
Hence here we have \(p\left ( x\right ) =2\cot \left ( 2x\right ) ,q\left ( x\right ) =4x\csc \left ( x\right ) \sec \left ( x\right ) ^{2}\). Therefore the integrating factor is
\begin{align*} \mu & =e^{\int p\left ( x\right ) dx}\\ & =e^{\int 2\cot \left ( 2x\right ) dx}\\ & =e^{-\frac {1}{2}\ln \left ( 1+\cot ^{2}\left ( 2x\right ) \right ) }\\ & =\frac {1}{\sqrt {1+\cot ^{2}\left ( 2x\right ) }}\end{align*}
Then the ode becomes
\begin{align*} \frac {d}{dx}\left ( y\mu \right ) & =\mu 4x\csc \left ( x\right ) \sec ^{2}\left ( x\right ) \\ \frac {d}{dx}\left ( y\frac {1}{\sqrt {1+\cot ^{2}\left ( 2x\right ) }}\right ) & =\frac {1}{\sqrt {1+\cot ^{2}\left ( 2x\right ) }}4x\csc \left ( x\right ) \sec ^{2}\left ( x\right ) \\ \frac {y}{\sqrt {1+\cot ^{2}\left ( 2x\right ) }} & =\int \frac {4x\csc \left ( x\right ) \sec ^{2}\left ( x\right ) }{\sqrt {1+\cot ^{2}\left ( 2x\right ) }}dx+c_{1}\\ y & =\sqrt {1+\cot ^{2}\left ( 2x\right ) }c_{1}+\sqrt {1+\cot ^{2}\left ( 2x\right ) }\int \frac {4x\csc \left ( x\right ) \sec ^{2}\left ( x\right ) }{\sqrt {1+\cot ^{2}\left ( 2x\right ) }}dx \end{align*}