Question

The general solution of the differential equation $\frac{dy}{dx}=\frac{y}{x}$ is
(a) log y = kx

(b) y = kx

(c) xy = k

(d) y = k log x

Answer 1

(b) y = kx

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