x \(\frac{dy}{dx}\) = x + y
⇒ \(\frac{dy}{dx}\) = \(\frac{x+y}x\) = 1 + y/x.
Let y/x = v
⇒ y = v x
⇒ \(\frac{dy}{dx}\) = v + x \(\frac{dv}{dx}\)
Then, v + x \(\frac{dv}{dx}\) = 1 + v
⇒ d v = dx/x
⇒ v = ln x + ln c, where ln c is integral constant.
⇒ y/x = ln cx (∵ ln A + ln B = ln AB)
⇒ y = x ln cx which is a solution of given differential equation.