The solution of the differential equation xdy - ydx = 0 represents

The differential equation is $xdy - ydx = 0$xdy−ydx=0

Move $ydx$ydx to the right side:

$$xdy = ydx$$xdy=ydx

Divide both sides by $xy$xy:

$$\frac{dy}{y} = \frac{dx}{x}$$ydy​=xdx​

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Integrate both sides:

$$\int \frac{dy}{y} = \int \frac{dx}{x}$$∫ydy​=∫xdx​

$$\ln|y| = \ln|x| + C$$ln∣y∣=ln∣x∣+C

where $C$C is the constant of integration.

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$$\ln|y| - \ln|x| = C$$ln∣y∣−ln∣x∣=C

Using the logarithm property $\ln(a) - \ln(b) = \ln(a/b)$ln(a)−ln(b)=ln(a/b):

$$\ln\left|\frac{y}{x}\right| = C$$ln​xy​​=C

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Taking exponential on both sides:

$$\left|\frac{y}{x}\right| = e^C$$​xy​​=eC

Since $e^C$eC is a positive constant, let $e^C = k$eC=k where $k > 0$k>0:

$$\frac{y}{x} = \pm k$$xy​=±k

Let $\pm k = m$±k=m where $m$m is any constant:

$$\frac{y}{x} = m$$xy​=m

$$y = mx$$y=mx

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The equation $y = mx$y=mx represents a straight line with slope $m$m and $y$y-intercept $0$0.

Therefore, the solution represents a straight line passing through the origin.