Question

Show that the given differential equation is homogenous and solve them
$Y^{\prime }=\frac{x+y}{x}$

Answer 1

$\frac{dy}{dx}=\frac{x+y}{x}$

Let $x=ax,y=ay$

$\therefore \frac{dy}{dx}=\frac{ax+ay}{ax}=\frac{a(x+y)}{ax}=\frac{x+y}{x}$

$\therefore$ Homogeneous differential eq.

Noe let $y=\nu x\Rightarrow \frac{dy}{dx}=\nu +x\frac{d\nu }{dx}$

$\therefore \nu +x\frac{d\nu }{dx}=\frac{x+\nu x}{x}=1+\nu$

$\Rightarrow \nu +x\frac{d\nu }{dx}=1+\nu$

$\Rightarrow d\nu =\frac{dx}{x}$

$\Rightarrow \int d\nu =\int \frac{dx}{x}$

$\Rightarrow \nu =logx+c$

$\Rightarrow \frac{y}{x}=logx+c$

$\Rightarrow y=xlogx+cx$.