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I understand that mechanically, an explicit solution is of the form y(x). So explicit solutions are straightforward enough. The differential equation in question involves derivatives of an unknown function, and the explicit solution is simply that function. But what does it mean for an implicit solution to define an explicit solution?

From examples I've seen, it seems to mean, "If you differentiate the implicit solution, then the resultant differential equation won't contradict the differential equation for which you claim the implicit solution is a solution." Is that right? Is there a better way to conceptualize it?

Thanks!

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An implicit solution is simply a solution that is not explicit. If you prefer: in that case we know that a solution exists, but we don't know how to express it explicitly.

"Most" solutions of differential are of this type since we cannot solve "most" equations explicitly, although we do know that they often have solutions (from the existence theorems, in a similar manner to that in the implicit function theorem).

Example: we don't know how to solve say $x'=x^{10}+\sin x$ ("solve" means "find the solutions explicitly of"), but we know that it has solutions.

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    What do you mean when you say that "an implicit solution is simply a solution?" I suppose that I don't understand what it means to "be a solution" of a differential equation. – 2017-01-28
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    You must have some notes, surely there is the definition of "solution" there. It is always the first definition of any ODE's course. – 2017-01-28
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    Implicit solutions were introduced but not defined. And as I indicated in my question, I understand what an explicit solution is. That feels very natural. I'm just getting hung up on implicit solutions. I mean, don't you solve a differential equation for its unknown function? So the implicit solution implicitly defines the unknown function, I presume. But what does it mean for an equation to implicitly define a function? – 2017-01-28
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    What does it mean to be "introduced" but not "defined"? As I said: you need to look for the notion of "solution" in your notes. There is no notion of "implicit solution", but instead what is in my answer (after you recall the notion of "solution" please read my answer again). – 2017-01-28
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    "Solution" was not defined either. – 2017-01-28
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    @JohnDoeVsJoeSchmoe A solution is simply "something" satisfying the equation, it has different meanings depending on context. For examples, when we solve $x - 1 = 0$ over $\mathbb{R}$, we try to find $x_0 \in \mathbb{R}$ such that $x_0 - 1 = 0$. Now we observe that $1 - 1 = 0$ and $1 \in \mathbb{R}$, so $1$ is a solution. Similarly, when we solve $y' - 1 = 0$ for $\mathbb{R} \to \mathbb{R}$ functions, we try to find $y_0: \mathbb{R} \to \mathbb{R}$ such that $y_0' - 1 = 0$. We observe that $x' - 1 = 1 - 1 = 0$ and $x$ is a function from $\mathbb{R} \to \mathbb{R}$, so $x$ is a solution. – 2017-11-28
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    Are implicit solutins "true" solutions? – 2018-10-23