Question

x cos² y dx = y cos² x dy

Answer 1

We have,

x cos² y dx = y cos² x dy

\[\Rightarrow y \sec^2 y dy = x \sec^2 x dx\]

Integrating both sides, we get

\[\Rightarrow y\int \sec^2 ydy - \int\left( \frac{dy}{dy} \times \int \sec^2 y dy \right)dy = x\int \sec^2 x dx - \int\left( \frac{dx}{dx} \times \int \sec^2 x dx \right)dx\]

\[ \Rightarrow y \tan y - \int\tan y dy = x \tan x - \int\tan x dx - C\]

\[ \Rightarrow y \tan y - \log \left| \sec y \right| = x \tan x - \log \left| \sec x \right| - C\]

\[ \Rightarrow x \tan x - y \tan y = \log\left| \sec x \right| - \log\left| \sec y \right| + C\]