Mathematics - Differential Equation Question with Solution | TestHub

Given, $\sin 2x\left(\frac{dy}{dx}-\sqrt{\tan x}\right)-y=0$ or, $\frac{dy}{dx}=\frac{y}{\sin 2x}+\sqrt{\tan x}$ or, $\frac{dy}{dx}-y\mathrm{cosec}2x=\sqrt{\tan x}$ Now, integrating factor (I.F) $=e^{\int -\mathrm{cosec}2x}$ or, I.F $=e^{-\frac{1}{2}\log |\tan x|}=e^{\log (\sqrt{\tan x}{)}^{-1}}$ $=\frac{1}{\sqrt{\tan x}}=\sqrt{\cot x}$ Now, general solution of eq. (1) is written as $$\begin{array}{rl}& \text{ y (I. F.) }=\int \mathrm{Q}(\text{ I.F.) }dx+c\\ & \therefore y\sqrt{\cot x}=\int \sqrt{\tan x}\cdot \sqrt{\cot x}dx+c\\ & \therefore y\sqrt{\cot x}=\int 1.dx+c\end{array}$$