I am supposed to prove that if $f(x,y)$ and $g(x,y)$ are homogeneous functions of degree $m$ and $n$ respectively and that $h=f+g$, such that $xh_{x}+yh_{y}=0$, then for some scalar $\alpha\in\mathbb{R}$, $f=\alpha g$.
Here is what I've tried so far. I've made use of Euler's theorem for homogeneous functions to write the following equations, from which I deduce that $f+g$ is also a homogeneous function of degree either $m$ or $n$, so clearly, whatever the function, $g$ it has to be a scalar multiple of $f$ and vice-versa. Is this reasoning correct and formal? Any inputs are appreciated. Thanks.
For $u(x,y)$ we have $${d\over dt}u(tx,ty)\Big |_{t=1}= [x\,u_x(tx,ty)+y\,u(tx,ty)]\Big |_{t=1}=xu_x+yu_y$$ Moreover, if $u$ is homogeneous of degree $k,$ then $${d\over dt}u(tx,ty)\Big |_{t=1}={d\over dt}t^k\Big |_{t=1}\ u(x,y)=k\,u(x,y)$$ i.e. $xu_x+yu_y=ku.$
The assumptions imply $$0=xh_x+yh_y=[xf_x+yf_y]+[xg_x+yg_y]=mf+ng.$$