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