How to express some coefficient of a generating function using only its derivatives

Question

I have a series of terms, $a_{1},a_{2},...,a_{k-1}$ that depend on $t$. I describe them in the form of the generating function, $Q(x,t)=\sum_{i=1}^{k-1}a_{i}(t)x^{i}$. My goal is to end up with a partial differential equation in $Q,Q_{t},Q_{x},x$, where subscripts indicate partial derivatives. My problem is that in calculating $Q_{t}$, I end up with terms in $a_{1}$ and $a_{2}$. For those interested, the equation I get is $$\begin{align} Q_{t}+\left(x-\frac{1+p}{2}\right)Q_{x}= \ &\frac{1-p}{2}(x+x^{k-1})(Q_{x}|_{1})\\ &-\left(\frac{1+p}{2}+\frac{1-p}{2}x+px^{k-1}\right)a_{1}\\ &+2pa_{2}, \end{align}$$ where $Q_{x}|_{1}$ is the partial $x$ derivative of $Q$ at $x=1$, and $p$ is a constant. With that in mind, could someone please explain to me how to express $a_{1}$ and $a_{2}$ in terms of $Q$ and its $x$ derivatives, if it is even possible? EDIT: Answered in comments