Question

The solution of the differential equation \( \sec^2 x \, \tan y \, dx + \sec y \, \tan x \, dy = 0 \) is:

• A. \( \tan x \, \tan y = C \)
• B. \( \tan y = \tan x = C \)
• C. \( \tan^2 x = C \)
• D. None of these

Hint:

When solving differential equations, try to separate variables and integrate both sides to find the general solution.

Answer 1

The Correct Option is A

We are given the differential equation: \[ \sec^2 x \, \tan y \, dx + \sec y \, \tan x \, dy = 0 \] Step 1: Rearrange the equation Rearrange the terms: \[ \frac{\sec y \, \tan x}{\sec^2 x \, \tan y} = -\frac{dy}{dx} \] Simplify: \[ \frac{\sec y \, \tan x}{\tan y \, \sec^2 x} = -\frac{dy}{dx} \] Step 2: Solve the equation Integrating both sides, we get the solution: \[ \tan x \, \tan y = C \] Thus, the correct answer is \( \tan x \, \tan y = C \).