the existence question
Before you try to find a solution to a differential equation, there is a quieter, more basic question worth asking: is there a solution at all? You are given a rule for the rate of change, say y' = f(x, y), together with a starting value y(x0) = y0, and you want to know whether some honest function actually threads through that starting point obeying the rule. It is easy to forget this can fail or be ambiguous, because the simple textbook equations almost always behave. But writing down an equation does not by itself conjure a function that solves it.
Concretely, an initial value problem asks for a function y(x), defined at least on some little interval around x0, that passes through the point (x0, y0) and whose slope at every point equals f(x, y) there. The existence question asks only whether at least one such function exists near x0 — not whether it is unique, and not how far it extends. The honest answer depends on how well-behaved f is: mere continuity of f already buys you a local solution (Peano's theorem), while stronger smoothness buys you more. A slope field is a good mental picture: existence is the claim that you can always follow the little arrows away from the starting point without the field tearing or disappearing.
This matters because it draws a firm line between three separate things people sloppily lump together: existence (is there a solution?), uniqueness (is there only one?), and global behaviour (how far does it last?). Each can hold or fail on its own. There are problems with a solution that is not unique, and problems whose unique solution exists only on a tiny interval and then blows up. Keeping the questions apart is the whole point of this part of the subject, and it is why the existence theorems are stated so carefully.
For y' = y with y(0) = 1, the function y = e^x exists, sails through (0, 1), and has slope equal to its own value everywhere — existence is obvious. For y' = y^(2/3) with y(0) = 0, a solution also exists (in fact y = 0 works), so existence is fine here too; what fails later is uniqueness.
Existence is the easy half: even the troublesome y^(2/3) example has a solution. The trouble is whether it is the only one.
Existence alone says nothing about uniqueness or how long the solution lasts — those are genuinely separate questions, each with its own theorem.