I'm trying to prove that the following PDE has no solutions that are non-constant functions $f\in C^1(\mathbb{R}^2)$.

$$
(y+1)u\_y + x u\_x = 0
$$

I'm not sure what is the theory that would help start formulating a solution for this since the material provided in my class is relatively bad explained and/or I'm missing something.

I tried first solving it with the method of characteristics:

$$
\frac{dy}{dx} = \frac{y+1}{x} \implies \frac{y+1}{x} = K
$$

Then I consider the variable change:
$$
\xi = \frac{y+1}{x}\\
\eta = y
$$
And I get of course:
$$
(\eta+1)u\_\eta = 0 \implies u=constant
$$

But I suspect this doesn not constitute the proof requested. This would just proof there is constants solutions.
I've tried also asumming it has such solution $u(x, y)$ non-constant to get to an absurd.

Then $u\_x \neq 0$ and $u\_y \neq 0$.

And Then it would be 

$$(y+1)u\_y = -xu\_x \quad \text{with} \quad (y+1)u\_y\neq 0; xu\_x\neq 0; y\neq1; x\neq0$$

But I don't get anything.