Find the general solution of the differential equation \(ydx = \left( {y - x} \right)dy\)

Concept:

The standard form of a linear equation of the first order is given by \(\frac{{dy}}{{dx}} + Py = Q\) where P, Q are arbitrary function of x.

The integrating factor of the linear equation is given by \(I.F. = {e^{\smallint pdx}}\)

The solution of the linear equation is given by \(y\left( {I.F.} \right) = \smallint Q\left( {I.F.} \right)dx + c.\)

Calculation:

\(ydx = \left( {y - x} \right)dy\)

\(y\frac{{dx}}{{dy}} = y - x\)

\(\frac{{dx}}{{dy}} + \frac{x}{y} = 1\)

It is form of \(\frac{{dx}}{{dy}} + Px = Q\)

\(I.F. = {e^{\smallint pdy}}\)

\(I.F. = {e^{lny}} = y\)

The solution of the linear equation is given by

\(x\left( {I.F.} \right) = \smallint Q\left( {I.F.} \right)dy + c.\)

\(x\left( y \right) = \smallint 1\left( y \right)dy + c\)

\(xy = \frac{{{y^2}}}{2} + c\)