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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 $C$:
$xy = C$
Correct Answer: xy=C