The general solution of the differential equation \(\rm \frac{ydx-xdy}{x} = 0\) is

Concept:

If the coefficient of \(\rm dx\) is only function of x and coefficient of \(\rm dy\) is only a function of y in the given differential equation then we can separate both \(\rm dx\) and \(\rm dy\) terms and integrate both separately.

\(\rm ⇒ \smallint f\left( x \right)dx = \smallint g\left( y \right)dy\)

Calculation:

To Find: Solution of the differential equation

\(\rm \frac{ydx-xdy}{x} = 0\)

⇒ ydx - xdy = 0

⇒ ydx = xdy 

⇒ \(\rm \frac{dy}{y}=\frac{dx}{x}\)

Integrating both sides, we get

\(\rm \int \frac{dy}{y}=\int \frac{dx}{x}\\⇒ \ln y = \ln x + \ln c\)   

\(\rm ⇒ \ln (y) = \ln cx\)                   (∵ ln x + ln y = ln (xy))

⇒ y = cx 

∴ y - cx = 0