Question

Show that the given differential equation is homogeneous and then solve it.

$y^{\prime }=\frac{x+y}{x}$

Answer 1

Given, $y^{\prime }=\frac{dy}{dx}=\frac{x+y}{x}$ ....(i)
Here, the given differential equation is homogeneous.
So, put $y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$
On putting values of $\frac{dy}{dx}$ and y in Eq.(i), we get
$\therefore v+x\frac{dv}{dx}=\frac{x+vx}{x}\Rightarrow v+x\frac{dv}{dx}=1+v\Rightarrow \frac{dv}{dx}=\frac{1}{x}\Rightarrow dv=\frac{dx}{x}$
On integrating both sides, we get
$\int dv=\int \frac{1}{x}dx\Rightarrow v=log|x|+C\Rightarrow \frac{y}{x}=log|x|+C$ $[\because v=y/x]$
$\Rightarrow y=xlog|x|+Cx.$
This is the required solution of the given differential equation.