The expectation has a well defined definition for both discrete and continuous random variables for E[g(X)] for any function g(X).
These types of equations are called Stochastic Differential Equations or SDE's for short.
If you are taking the expectation with respect to X and t is deterministic, then you can calculate E[g(X,t)] where the g(X,t) is inside the expectation if you have the PDF for X.
I'm speculating that t is not a random variable, but please inform us if that is not correct.
That's basically all there is to it.
Even if you can't get an analytic solution easily, you can use one of the many numerical packages to get a numeric solution in terms of a function of your time variable t (since t is a variable independent from X and is not a random variable) and then you will get some function in terms of t.
Now since you mention that this is an implicit form, you will have to use the use the chain rule of the Fundamental Theorem of Calculus to get rid of the integral.
The Integral for the expectation will look like E[g(X,t)] = Integral (-infinity to infinity) g(X,t)*f(x)dx where f(x) is the PDF for the random variable X (assuming your expectation is with respect to the random variable X).