Question

Show that the differential equation $\left(x-y\right)\frac{dy}{dx}=x+2y$ is homogeneous.

Answer 1

A differential equation of the form $\frac{dy}{dx}=F(x,y)$ is said to be homogenous if $F(x,y)$ is a homogeneous function of $degree=0$

Given that

$(x-y)\frac{dy}{dx}=x+2y$

We can rewrite the above equation as

$\Rightarrow \frac{dy}{dx}=\frac{x+2y}{x-y}$

$\Rightarrow F(x,y)=\frac{x+2y}{x-y}$

$\Rightarrow F(kx,ky)=\frac{kx+2ky}{kx-ky}$

$\Rightarrow F(kx,ky)=\frac{k(x+2y)}{k(x-y)}$

$\Rightarrow F(kx,ky)=\frac{(x+2y)}{(x-y)}=k^{0}F(x,y)$

Hence this a homogeneous function with $degree=0$