Question

Solve the following homogeneous differential equation : \[ x \frac{d y}{d x}=x+y \]

Answer 1

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.