Question

The general solution of the differential equation $\frac{dx}{y}+\frac{dy}{y}$ = 0 is ________________.

Answer 1

Given: $\frac{dx}{y}+\frac{dy}{y}$ = 0


$$\begin{gathered} \frac{dx}{y}+\frac{dy}{y}=0 \\ \Rightarrow \frac{dx}{y}=-\frac{dy}{y} \\ \Rightarrow dx=-y\frac{dy}{y} \\ \Rightarrow dx=-dy \\ \Rightarrow dy=-dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow \int dy=-\int dx \\ \Rightarrow y=-x+C,\mathrm{where}C\mathrm{is}\mathrm{arbitrary}\mathrm{constant} \end{gathered}$$


Hence, the general solution of the differential equation $\frac{dx}{y}+\frac{dy}{y}$ = 0 is $\underline{y=-x+C}$.