Q94 / 180 · KCET 2016 · Mathematics
The solution for the differential equation \( \frac{d y}{y}+\frac{d x}{x}=0 \)
- A\( \frac{1}{y}+\frac{1}{x}=C \)
- B\( \log x \cdot \log y=c \)
- Correct answerC\( x y=c \)
- D\( x+y=c \)
Solution
Given differential equation is,
\( \frac{d y}{y}+\frac{d x}{x}=0 \)
Integrating both the sides
\[ \begin{array}{l} \int \frac{d y}{y}+\int \frac{d x}{x}=\int \operatorname{Od} x \\ \Rightarrow \log y+\log x=\log c \\ \Rightarrow \log (x y)=\log c \\ \Rightarrow x y=c \end{array} \]
\( \frac{d y}{y}+\frac{d x}{x}=0 \)
Integrating both the sides
\[ \begin{array}{l} \int \frac{d y}{y}+\int \frac{d x}{x}=\int \operatorname{Od} x \\ \Rightarrow \log y+\log x=\log c \\ \Rightarrow \log (x y)=\log c \\ \Rightarrow x y=c \end{array} \]
Hence, the correct answer is option: 'C'.