What exacly does solving a differential equation mean? I am taking an introductory differential equation course.
Lots of questions in the textbook (and all textbooks I have found on differential equations) are like:
Solve $y'(x) = f(x)$.
What does it mean? What should be $y$'s domain? Should I find $y$ that satisfies this at all inputs of its domain? Should I find $y$ regardless of its domain that satisfies this at some inputs?
Why are all questiosn regarding differential equations in almost all textbooks at this level are like that? Why can't they specify the kind of y?
Furthurmore, consider a simple differential equation question: "Solve $y' = 0$." $y(x)= x^2$ satisfies this at $x = 0$. $\sin (x)$ satisfies this at $x$ such that $\cos (x) =0$. $y: (-2,2) \to \mathbb R, x \mapsto 0 $ satisfies this at every point in its domain. Finally,$y:\mathbb R\to\mathbb R \cup\text{the set of 2 by 2 matrices},x\mapsto0\text{ if }x <0,x\mapsto\begin{bmatrix}0&0\\0&0\end{bmatrix}\text{ if }x\geq0$. It satisfies this on $(-\infty, 0)$.
From the examples above, it is clearly no way I can describe all such functions. But clearly, they are all solutions in that they satisfy the equation at some input(s).
What I have been doing in cases where the question does not specify the domain of $y$ is that I assume $y$’s domain is a subset of $\mathbb R$, and codomain is $\mathbb R$, and $y$ satisfies the differential equation on an open interval which is a subset of the domain of $y$; and I try to find the behavior of $y$ only on that open interval. The reason for this is that this task is relative easier, and most modeling questions involve a similar task like this.
But it is still quite uncomfortable having to solving homework and exams problems like that without having a clear instruction.
The professor isn't very helpful. He keeps saying $y=C$ is the general solution and sometimes you need a particular solution. I don't know whether he does not understand my question or deliberately insisting on not revealling details.
 A: Q:  What does it mean to solve a differential equation?A:  It means finding a function such as $y(x)$ that solves the given equation, that is, where you take the appropriate derivates, add and multiply, and you have an identity.
Q:  What does it mean for a function to [be] a solution to a differential equation?A:  It means if you take the derivatives of your solution function, add, multiply, and so on according to the equation and you get an identity... that is, the values are always the same on both sides of the equal sign.
Q:  If a function's domain is not an open interval, but one or more open interval(s), is it not a solution to the differential equation even though it satisfies the given differential equation at every point in its domain?A:  Finding a solution often means finding the domain where a solution is valid.  There may be some domains, or points, where some portion of the equation is not defined.
A: Your question is an excellent question, one that I have been asking myself for a very long time. Really, why is it that the concept of solving equations is taught in the way that it is taught, and not in a way that makes it precise what exactly you are looking for? The reason is that primary education, even at the undergraduate level, teach mathematics, not so that you understand them conceptually, but so that you can get good at computational techniques and symbolic manipulation. In my personal view, this is the wrong way to teach mathematics, but regardless, that is how it is, and so that is how education is structured in most countries in the world. That is why teachers, textbooks, and educational resources on the Internet fail to really provide an explanation for what it means to solve an equation.
So, what does it mean to solve an equation? Every equation can ultimately be boiled down to writing $f(x)=y,$ where $f$ is a function $X\to{Y}$ (in other words, the domain is $X,$ and the codomain is $Y$), $x$ is some object in the set $X,$ and $y$ is some object in the set $y.$ The idea is to find which objects in the set $X$ make the equation $f(x)=y.$ That is all it means.
But, there is a problem. Very often, the equation you are given gives you an explicit expression for $f,$ but fails to specify the domain of $f.$ As such, it makes it ambiguous as to what you are supposed to be looking for. Consider for example the equation $x+x=0.$ What are the solutions to this equation? You may think it is easy to say what the solutions are: obviously $x=0,$ and nothing else. But you would be wrong. If I have an algebraic structure where $0+0=0,$ $1+0=0+1=1,$ and $1+1=0,$ then $x+x=0$ implies $x=0$ or $x=1.$ Is this a perfectly valid answer? Yes, because you never specified the domain $X,$ you just gave me what is effectively a meaningless string of symbols, and told me to fill in the blank. Now, if you tell me that the domain is $\mathbb{R},$ then suddenly, that changes things. Specifying the domain makes it so that the set of solutions is well-defined: obviously, it is some subset of $\mathbb{R},$ and it is unique. Here is a more extreme example: consider the equation $x\cdot{x}=1.$ How many solutions does it have? Well, that depends on the domain. If the domain is $\mathbb{N},$ then it has $1$ solution. If the domain is $\mathbb{R},$ then it has two solutions. But if the domain is the set of $2\times2$ matrices, then there are infinitely many solutions. All three sets can be equipped with an algebraic structure where the concept of multiplication, denoted $\cdot,$ is well-defined. Since I have specified a symbol $\cdot,$ but not the domain of the function it represents, I have not uniquely specified that function, and so asking to solve the equation is more or less nonsense. A well-written question will always ask something along the lines of "find all $x\in{S}$ such that $x\cdot{x}=1$." Specifying $S$ is what makes this a well-written question.
Differential equations are no different. a basic first order differential equation looks like this: $$F(x,f(x),D[f](x))=0,$$ but this is not a well-written question to answer, because the domain of $F$ has not been specified, and the domain of $f$ has not been specified. Even the simplest equation $D[f]=g$ is ambiguous to solve. Why? Because I know that $D$ stands for the derivative operator, but the specific class of differentiable functions that would be the domain of $D$ has not been specified. Am I being asked for all the functions differentiable in $(0,1)$? Differentiable on $(0,\infty)$? Or differentiable over all of $\mathbb{R}$? A well-written question to solve a differential equation looks like this: "find all $y\in{C(\mathbb{R})}$ such that $D[y]=f$," where $C(\mathbb{R})$ is the set of all functions $\mathbb{R}\to\mathbb{R}$ that are differentiable everywhere. And the equation $D[y]=f$ is definitionally equivalent to $y'(x)=f(x)$ for all $x\in\mathbb{R}$ in that case. Thus there is no ambiguity, as long as $f$ is well-defined. And in higher-level settings, you will find that questions are better written and unambiguous. But if you are just being introduced to the topic of differential equations, then it will usually be the case that texts and professors will be very sloppy. They will think that "the domain is implied from context" or something like that, and they will think that even in the situations where there actually is no context at all or it is very much not implied. In those situations, I advise that you ask questions and inquire further, so that you can get an idea for what the text actually means or what your professor actually means when he says anything. Ultimately, this all boils down to communication.
But, with that being said, most ordinary differential equations we are interested in solving are equations where the unknown function has domain $\mathbb{R}$ and are differentiable everywhere in the domain, and satisfy the equation everywhere in the domain. After all, there really is not much utility to an equation that is only satisfied in some limited part of the domain. In those cases, we just restrict the domain, and state it explciitly. Again, this is a matter of communication, and communication often comes with conventions. though I think those conventions are misused more often than not.
