4.5.1.3 Example 3 $y^{\prime }+2y\cot \left(2x\right)=4x\csc \left(x\right){\sec }^2\left(x\right)$

$$y^{\prime }+2y\cot \left(2x\right)=4x\csc \left(x\right){\sec }^2\left(x\right)$$

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

$$\begin{array}{rl}\mu & =e^{\int p\left(x\right)dx}\\ & =e^{\int 2\cot \left(2x\right)dx}\\ & =e^{-\frac{1}{2}\ln (1+{\cot }^2\left(2x\right))}\\ & =\frac{1}{\sqrt{1+{\cot }^2\left(2x\right)}}\end{array}$$

Then the ode becomes

$$\begin{array}{rl}\frac{d}{dx}\left(y\mu \right)& =\mu 4x\csc \left(x\right){\sec }^2\left(x\right)\\ \frac{d}{dx}\left(y\frac{1}{\sqrt{1+{\cot }^2\left(2x\right)}}\right)& =\frac{1}{\sqrt{1+{\cot }^2\left(2x\right)}}4x\csc \left(x\right){\sec }^2\left(x\right)\\ \frac{y}{\sqrt{1+{\cot }^2\left(2x\right)}}& =\int \frac{4x\csc \left(x\right){\sec }^2\left(x\right)}{\sqrt{1+{\cot }^2\left(2x\right)}}dx+c_1\\ y& =\sqrt{1+{\cot }^2\left(2x\right)}{c}_1+\sqrt{1+{\cot }^2\left(2x\right)}\int \frac{4x\csc \left(x\right){\sec }^2\left(x\right)}{\sqrt{1+{\cot }^2\left(2x\right)}}dx\end{array}$$