Question

The solution of \(\dfrac{dx}{x}+\dfrac{dy}{y}=0\) is

• A. \(x=ky\)
• B. \(\dfrac{1}{x}+\dfrac{1}{y}=k\)
• C. \(x+y=k\)
• D. \(xy=k\)

Hint:

\(\ln A+\ln B=\ln(AB)\). Replace \(e^{C}\) by a new constant \(k\).

Answer 1

The Correct Option is D

Idea. Integrate each fraction; logarithms add. Exponentiate to remove \(\ln\).
\[ \ln|x|+\ln|y|=C\Rightarrow \ln|xy|=C\Rightarrow xy=e^{C}=k. \]