Hello. The first equation is a differential equation about the variable t, which shows that there must be a function T(t) about t in the solution.

The second equation is a differential equation about the variable x, which shows that there must be a function X(x) about x in the solution.

So we can know that the form of ψ(x, t) must be a binary function with separable variables about x and t:

ψ(x，t)=X(x)T(t)

ψ(x, t) is acted by operator H, but actually acts on x (X(X)；; But when ψ(x, t) is acted by operator ih, it actually acts on T(t). Please refer to the separation of variables method in mathematical physics methods. The example of standing wave motion of string mentioned in it is exactly the same as the formal principle of determining ψ(x, t) here.