Question

The general solution of the differential equation \[\frac{dy}{dx} = \frac{y}{x}\] is

Options

• log y = kx

• y = kx

• xy = k

• y = k log x

Answer 1

y = kx

 

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