Let $X$ be a PDE. If we apply the method of characteristics to $X\cdot y_1$(x)$=1$, why do we get $\frac{dx_1}{...}=\frac{dx_2}{...}=\frac{dy_1}{1}$

Question

Being more specific I'm talking about that X:

$X=x_1\:\frac{\partial }{\partial x_1}\:+\:2x_{2\:}\frac{\partial }{\partial x_2}$

If we apply the method of characteristics to $X\cdot y_1=1$, we get $\frac{dx_1}{x_1}=\frac{dx_2}{2x_2}=\frac{dy_1}{1}$

Why? Our PDE is

$x_1\:\frac{\partial \:y_1}{\partial \:x_1}\:+\:2x_{2\:}\frac{\partial y_1\:}{\partial \:x_2}=y_1$

Shouldn't we then get:

$\frac{dx_1}{x_1}=\frac{dx_2}{2x_2}=\frac{dy_1}{y_1}$