Mathematics - Differential Equation Question with Solution | TestHub

Given that $$\begin{array}{rl}& \left(\frac{dy}{dx}\right)\tan x=y{\sec }^{2}x+\sin x\\ & \tan x\left(\frac{dy}{dx}\right)-y{\sec }^{2}x=\sin x\\ & \frac{dy}{dx}+\left(\frac{-{\sec }^{2}x}{\tan x}\right)y=\frac{\sin x}{\tan x}\\ & \text{ Here, }P=-\frac{{\sec }^{2}x}{\tan x}\text{ and }Q=\cos x\\ & \text{ IF }=e^{\int Pdx}=e^{\int \frac{-{\sec }^{2}x}{\tan x}dx}\\ & \text{ IF }=e^{-\log (\tan x)}=e^{\log \cot x}=\cot x\\ & \text{ Now, }y\times \mathrm{IF}=\int (Q\times \mathrm{IF})dx+C\\ & y\cdot \cot x=\int \cos x\cdot \cot x+C\\ & y\cdot \cot x=\int \frac{1-{\sin }^{2}x}{\sin x}+C\\ & y\cdot \cot x=\log |\mathrm{cosec}x-\cot x|+\cos x+C\end{array}$$