I have a formula that generates PDEs: $$ \partial_t^{k_0}g(t)\partial_x \partial_t^{n-k_0}F(x,t)\frac{(n-1)!}{(n-k_0)!(k_0+1)!}+\partial_t^nF(x,t)=0 $$ I don't think any other information is necessary except that $g(t):\mathbb{R}\rightarrow\mathbb{R}$, $F(x,t):\mathbb{R}^2\rightarrow\mathbb{R}$ and $n, k_0\in\mathbb{N}^*$, with $n>k_0$, $k_0$ being fixed. (I'm giving the equation only to describe the problem)
I'm interested in methods of showing if some function $F$ satisfies the PDE for $n=k_0+1$, it will also satisfy the PDEs $\forall n>k_0+1$.
I am not sure if this really is true or not, I just tried to compute for simple cases and prove this using brute force and I got stuck. Getting rid of $\partial_t^{k_0}g(t)$ for 2 values of $n$ did not help me too much so I thought there must be other methods that I am not aware of for proving that 2 or more PDEs have the same solutions. And here comes the question:
Are they?