When we write $y'' + y' + y = x$, for a function $y$, do we really mean $y''(x) + y'(x) + y(x) = x$? I have recently started exploring differential equations at a graduate level (partial and ordinary) and really started thinking about why we write them the way we do. Let (since this seems to be the common symbol of choice) $y: \mathbb{R} \to \mathbb{R}$ such that $x \mapsto y(x)$. Just to be fussy, we will consider $y \in C^{\infty}(\mathbb{R})$. Now, for example, consider the second order differential equation $$y'' + y' + y = x \tag{i}$$ My question is in regards to this notation: Why do we write the function and it's derivatives this way in the equation rather than writing: $$y''(x) + y'(x) + y(x) = x \tag{ii}$$ using the function values? When we write (i), do we really mean (ii)? This may be a bit pedantic, but I just want to be thorough.
 A: The context of the equation
\begin{equation}
y^{\prime\prime} + y^{\prime} + y = x
\end{equation}
means that given a function described by the following proposition:
\begin{equation}
\forall x \in \mathbb{R}, y^{\prime\prime}\left(x\right) + y^{\prime}\left(x\right) + y\left(x\right) = x,
\end{equation}
use deduction rules to deduce a new proposition in the following form:
\begin{equation}
\forall x \in \mathbb{R}, y\left(x\right) = ...
\end{equation}
Of course, rigorously speaking, you should first prove the existence and uniqueness of such a function $y$:
\begin{equation}
\exists! y: \mathbb{R} \mapsto \mathbb{R}, \forall x \in \mathbb{R}, y^{\prime\prime}\left(x\right) + y^{\prime}\left(x\right) + y\left(x\right) = x,
\end{equation}
which cannot be the case without boundary values. Thus, to specify a unique solution to your equation, you need to provide boundary values.
In the end, you conclude the following proposition as a solution to the problem:
\begin{equation}
\forall y:\mathbb{R} \mapsto \mathbb{R}, \left[\forall x \in \mathbb{R}, y^{\prime\prime}\left(x\right) + y^{\prime}\left(x\right) + y\left(x\right) = x \wedge y\left(0\right) = ... \wedge y^{\prime}\left(0\right) = ...\right] \longrightarrow \left[y\left(x\right) = ...\right]
\end{equation}
It should be noted that providing an analytical solution is not the only way to solve a differential equation (ordinary or partial). From set theory, functions are sets of 2-tuples. As long as you have a way to specify each 2-tuple in a function, the differential equation is solved. In this context, numerical approaches are also a convenient way to specify 2-tuples of a function.
A lot of times, people don't think about the logical meaning behind solving equations. Education of logic systems in math is often ignored in today's education system. Recently, someone even told me training in first order logic is not important to mathematicians. This is exactly the reason why math is so formidable to most people: using daily language to talk about a thing in the world of object language is not gonna lead to the success of most people. People fear math as math is not reachable to them. To make people truly understand the tool of math (they don't have to be masters using this tool for creative inventions), languages should be taught again from scratch.
