4. solve the given differential equation by using appropriate substitution. The DE is homogenous. x=√x = y +√√x²y²₂ + 30 X-

The given differential equation is , 

               $x\frac{dy}{dx} = y+\sqrt{x^2-y^2} ; x>0$

and the differential equation is homogeneous .

(.)  The first order differential equation $\frac{dy}{dx}= f(x,y)$ is said to homogeneous if , there exists        a function $g$ such that $f(x,y)$ can be expressed as $g\left(\frac{y}{x}\right)$ .

(.) If the first order differential equation is homogeneous then the substitution $y=vx$                     transforms the differential equation into a separable equation in $v$ and  $x$ .