Find the general solution of the differential equationy′=x−y+3x−y

A differential equation is an equation in x,y and derivatives of y. A first order differential equation is on in which the highest order derivative present in the differential equation is first order.
A differential equation in some cases can be simplified to a simpler differential equation with an appropriate substitution. For given differential equation find an appropriate substitution to simplify the differential equation.
Given differential equation is ${\displaystyle y^{\prime }=\frac{x-y+3}{x-y}}$. Use the substitution x-y=u. Differentiating this gives ${\displaystyle 1-\frac{dy}{dx}=\frac{du}{dx}}$. This can be rewritten as ${\displaystyle 1-\frac{du}{dx}=\frac{dy}{dx}}$. Substitute this in the differential equation and integrate.
${\displaystyle 1-\frac{du}{dx}=\frac{u+3}{u}}$
${\displaystyle 1-\frac{du}{dx}=1+\frac{3}{u}}$
${\displaystyle -\frac{du}{dx}=\frac{3}{u}}$
${\displaystyle -udu=3dx}$
${\displaystyle \frac{u^2}{2}=3x+C}$
${\displaystyle u^2=-6x-2C}$
${\displaystyle u=\sqrt{-6x-2C}}$
${\displaystyle x-y=\sqrt{-6x-2C}}$
${\displaystyle y=x-\sqrt{-6x-2C}}$
Hence, general solution to the differential equation is ${\displaystyle y=x-\sqrt{-6x-2C}}$