Question

The solution for the differential equation $\frac{dy}{y}+\frac{dx}{x}=0$ is:

• A. $\frac{1}{y}+\frac{1}{x}=c$
• B. $\log x.\log y=c$
• C. $xy=C$
• D. $x+y=c$

Answer 1

The correct option is C $xy=C$

We have, $\frac{dy}{y}+\frac{dx}{x}=0$

Integrate both sides,

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

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

, since differentiation any constant is 0

$\Rightarrow \ln x+\ln y=c$, use basic formula

$\Rightarrow \ln \left(xy\right)=c$, product rule of logarithm $\log a+\log b=\log \left(ab\right)$

$\Rightarrow xy=e^c=C$