Question

The general solution of the differential equation \[ \sec^2 x \tan y\,dx + \sec^2 y \tan x\,dy = 0 \] is

• A. \( \tan x \tan y = c \)
• B. \( \sec x \tan y = c \)
• C. \( \sec x \sec y = c \)
• D. \( \tan x \sec y = c \)

Hint:

When variables can be separated, look for expressions whose derivatives already appear in the equation.

Answer 1

The Correct Option is A

Step 1: Rearrange the given equation.
\[ \sec^2 x \tan y\,dx = -\sec^2 y \tan x\,dy \]

Step 2: Separate the variables.
\[ \frac{\sec^2 x}{\tan x}\,dx = -\frac{\sec^2 y}{\tan y}\,dy \]

Step 3: Integrate both sides.
\[ \int \frac{\sec^2 x}{\tan x}\,dx = \int -\frac{\sec^2 y}{\tan y}\,dy \] \[ \ln|\tan x| + \ln|\tan y| = C \]

Step 4: Combine logarithms.
\[ \ln|\tan x \tan y| = C \]

Step 5: Conclusion.
\[ \boxed{\tan x \tan y = c} \]