Question

Solve the differential equation: \[ \cos{x} \cos{y} \, dy - \sin{x} \sin{y} \, dx = 0 \]

Hint:

When solving separable differential equations, rearrange terms and integrate each side independently.

Answer 1

Rearrange the terms: \[ \cos{x} \cos{y} \, dy = \sin{x} \sin{y} \, dx \] Divide both sides by \( \cos{x} \cos{y} \) and \( \sin{x} \sin{y} \): \[ \frac{dy}{\sin{y}} = \frac{dx}{\cos{x}} \] Integrating both sides: \[ \int \frac{dy}{\sin{y}} = \int \frac{dx}{\cos{x}} \] The integral of \( \frac{1}{\sin{y}} \) is \( \ln|\tan{\frac{y}{2}}| \), and the integral of \( \frac{1}{\cos{x}} \) is \( \ln|\sec{x}| \). Thus, we get: \[ \ln|\tan{\frac{y}{2}}| = \ln|\sec{x}| + C \] Exponentiate both sides to remove the logarithms: \[ |\tan{\frac{y}{2}}| = C_1 |\sec{x}| \] Finally, we can solve for \( y \), but this equation gives the general solution to the differential equation.