Question

The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is

Options

• xy = C

• x = Cy²

• y = Cx

• y = Cx²

Answer 1

y = Cx

 

We have,

\[\frac{y dx - x dy}{y} = 0\]

\[ \Rightarrow y dx = x dy\]

\[ \Rightarrow \frac{1}{y}dy = \frac{1}{x}dx\]

Integrating both sides, we get

\[\int\frac{1}{y}dy = \int\frac{1}{x}dx\]

\[ \Rightarrow \log y = \log x + D\]

\[ \Rightarrow \log y - \log x = \log C\]

\[ \Rightarrow \log\left( \frac{y}{x} \right) = \log C\]

\[ \Rightarrow \frac{y}{x} = C\]

\[ \Rightarrow y = Cx\]