Question

The general solution of the differential equation $\frac{ydx-xdy}{y}=0$, is
(a) xy = C
(b) x = Cy²
(c) y = Cx
(d) y = Cx²

Answer 1

(c) y = Cx


$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{ydx-xdy}{y}=0 \\ \Rightarrow ydx=xdy \\ \Rightarrow \frac{1}{y}dy=\frac{1}{x}dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \frac{1}{y}dy=\int \frac{1}{x}dx \\ \Rightarrow \log y=\log x+D \\ \Rightarrow \log y-\log x=\log C \\ \Rightarrow \log \left(\frac{y}{x}\right)=\log C \\ \Rightarrow \frac{y}{x}=C \\ \Rightarrow y=Cx \end{gathered}$$