Consider the set of algebraically primitive functions:
that is the set of functions who can be created through some recursive expression involving arithmetic operators. Example
$$y = x +1$$ $$y = x^2$$ $$y = 2^x$$ $$y = 3^{x + \sqrt{x} - \sqrt{2}} + \frac{2}{\sqrt{1 + x^2 + \log_5(x)}}$$
all these functions have the property of being based on arithmetic, algebra, and extensions of those.
Consider the Differential Equation
$$\frac{dy}{dx} = y$$
It is clear that the solution to this diff EQ cannot be expressed using any version of the functions above. Unless we create a number $e$ and consider the set of expressions above that can also have the number $e$. Then the solution:
$$C_1e^x$$
becomes accessible and can be used to solve this equation.
This process of defining new terms to solve differential equations is analogous to create ultra-radicals to solve high order polynomials. Is there some abstract algebraic way of looking at differential equations, determining if they are solvable given certain tools, and if not, what additional numbers/functions need to be defined to make them solvable?