Consider the equation
$$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}\Lambda(n)\Phi_n(x,t)$$
where $\Lambda(n)$ represents an arbitrary number theoretic function, that is a function $\Lambda:\Bbb N \to \Bbb C.$ So far I've solved it for $\Lambda(n)=2T_n$ for $T_n$ the triangular numbers. That is I found a $\Phi$ and a $\Lambda$ which satisfies the relation. See also this related question and answer.
Is there a name for PDE's like this? Are there any resources that discuss PDE's involving number theoretic functions?