Solve the differential equation ^2 y x d y+ ^2 x y d x=0 - Vidyadip

Question

Solve the differential equation $\sec ^2 y \tan x d y+\sec ^2 x \tan y d x=0$

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Answer

$\sec ^2 y \tan x d y+\sec ^2 x \tan y d x=0$
Dividing both sides by $\tan x \tan y$, we get
$ \frac{\sec ^2 y \tan x}{\tan x \tan y} d y+\frac{\sec ^2 x \tan y}{\tan x \tan y} d x=0$
$\therefore \frac{\sec ^2 x}{\tan x} d x+\frac{\sec ^2 y}{\tan y} d y=0 $
Integrating on both sides, we get
$ \int \frac{\sec ^2 x}{\tan x} d x+\int \frac{\sec ^2 y}{\tan y} d y=0$
$\therefore \log |\tan x |+\log |\tan y |=\log | c |$
$\therefore \log |\tan x \cdot \tan y |=\log | c |$
$\therefore \tan x \tan y = c $

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