Question

Solve the differential equation \( \cos x \cos y \, dy - \sin x \sin y \, dx = 0 \).

Hint:

Separate variables for differential equations; use standard integrals for trigonometric functions.

Answer 1

Rewrite: \[ \cos x \cos y \, dy = \sin x \sin y \, dx \quad \Rightarrow \quad \frac{\cos y}{\sin y} \, dy = \frac{\sin x}{\cos x} \, dx. \] \[ \cot y \, dy = \tan x \, dx. \] Integrate: \[ \int \cot y \, dy = \int \tan x \, dx. \] \[ \ln |\sin y| = -\ln |\cos x| + c \quad \Rightarrow \quad \ln |\sin y| + \ln |\cos x| = c \quad \Rightarrow \quad \sin y \cdot \cos x = C. \] Solution: \( \sin y \cos x = C \).