# Exponential Operator Representing Solution to Autonomous First Order Differential Equations

I am studying Dominic Edelen's Applied Exterior Calculus Section 1-4 as a start on understanding derivatives in differential geometry. He uses an exponential function containing a derivative operator as a way of "representing" the solution. My question is whether this is an expression that can actually be evaluated, or just somehow a non-evaluable symbol? If actually evaluable then how?

Here is a very simple single variable example. I hope that Mathematica notation is clear enough here.

differentialEquations = {x'[t] == 4 x[t] + 3, x[0] == 0};
dsol = DSolve[differentialEquations, x, t][[1, 1]];


This gives as a solution and a series expansion for the solution:

Now I can calculate the series coefficients recursive using the chain rule (and maybe that is all that is meant by the exponential operator expression). But Edelen gives the following expression as representing the solution:

where (4x+3) is the right hand side of the differential equation and the operator part is the partial derivative with respect to x. I have not understood how this is to be evaluated. Is the x expression just to be pushed onto the derivative operator, giving 1, or is the whole expression to be first expanded somehow into a series expression - and then how is the derivative operator to be handled? Using the first method gives:

and is clearly incorrect.

So, can such exponential operator expressions be directly evaluated and if so how?