Question

\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x \log x}\]

Answer 1

We have, 
\[\frac{dy}{dx} - x \sin^2 x = \frac{1}{x\log x}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{x\log x} + x \sin^2 x\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{x\log x} + \frac{x}{2}\left( 1 - \cos 2x \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{x\log x} + \frac{x}{2} - \frac{x}{2}\cos 2x\]
\[ \Rightarrow dy = \left[ \frac{1}{x\log x} + \frac{x}{2} - \frac{x}{2}\cos 2x \right]dx\]
Integrating both sides, we get
\[\int dy = \int\left[ \frac{1}{x\log x} + \frac{x}{2} - \frac{x}{2}\cos 2x \right]dx\]
\[ \Rightarrow y = \int\frac{1}{x\log x}dx + \frac{1}{2}\int x dx - \frac{1}{2}\int\left( x \cos 2x \right)dx\]
\[ \Rightarrow y = \log\left| \log x \right| + \frac{1}{2} \times \frac{x^2}{2} - \frac{1}{2}\int x_I \times \cos_{II} 2x dx \]
\[ \Rightarrow y = \log\left| \log x \right| + \frac{x^2}{4} - \frac{x}{2}\int\left( \cos 2x \right)dx + \frac{1}{2}\int\left[ \frac{d}{dx}\left( x \right)\int\left( \cos 2x \right) dx \right]dx\]
\[ \Rightarrow y = \log\left| \log x \right| + \frac{x^2}{4} - \frac{x\sin 2x}{4} - \frac{\cos 2x}{8} + C\]
\[\text{ Hence, }y = \log\left| \log x \right| + \frac{x^2}{4} - \frac{x\sin 2x}{4} - \frac{\cos 2x}{8} +\text{ C is the solution to the given differential equation.}\]