Given the differential equation: $\frac{dx}{x} + \frac{dy}{y} = 0$

Integrate both sides:

$\int \frac{dx}{x} + \int \frac{dy}{y} = \int 0$

$\log |x| + \log |y| = C'$ where $C'$ is the constant of integration.

Using the logarithm property $\log a + \log b = \log (ab)$:

$\log |xy| = C'$

Taking the exponential of both sides:

$|xy| = e^{C'}$

Since $e^{C'}$ is a constant, we can replace it with another constant $C$:

$xy = C$