I'm studying about the Hirota's direct method for PDE's and showing some properties of the bilinear $D-$operatos of Hirota. So, for 2 functions $a,b$, the $D-$operator is defined by $$D_{x}^{n}a\cdot b=\left(\partial_{x}-\partial_{y}\right)^{n}a(x)b(y)\mid_{y=x},\qquad n\in\mathbb{N}$$ and with this new operator, we define $D_{z}$ by $D_{z}=D_{t}+\varepsilon D_{x}$ where $\varepsilon$ is a parameter. What I want to show is that $D_{z}a\cdot b=0\Longleftrightarrow~ a=\lambda b$ for some constant $\lambda\in\mathbb{C}$. One way is easy, which is $\Leftarrow$, but I can't prove the other way.
What I've done is that for the definition for $D_{x}^{n}$ operator, I proved that $D_{x}a\cdot b=a_{x}b-ab_{x}$ and substituting it and assuming $D_{z}a\cdot b=0$ I just have $a_{t}b+\varepsilon a_{x}b=ab_{t}+\varepsilon ab_{x}$ and from here I'm stucked.