A differential equation, or an identity?

[etotheipi]

This is quite literally a showerthought; a differential equation is a statement that holds for all $x$ within a specified domain, e.g. $f''(x) + 5f'(x) + 6f(x) = 0$. So why is it called a differential equation, and not a differential identity? Perhaps because it only holds for a specific set of solutions for $f(x)$?

 

[S.G. Janssens]

Yes, I think you got it.

A solution $f$ (not $f(x)$) is defined as a twice continuously differentiable function (on the domain $J$ that you mentioned, typically an interval) that turns the equation into an identity on $J$.

 

[fresh_42]

Equations are normally named as such: equations. An identity is - if at all - used for certain fixed equations, or in physics when it relates different quantities. But then it's more often a law. The word identity is useless, better is formula.

If the above was called a differential identity, then this would suggest that it holds in a general framework. The expression for all $x$ in a certain domain is not quite true. It hides the variation of differential equations: they describe a family of motions (physics) or flows (mathematics), because without initial values specified, a differential equation has a family of solutions, aka flows. Selecting initial values means selecting a certain solution (flow) among them. Draw a vector field (attach little vectors at any point of your paper). Then flow through this vecorfield along the littles arrows for a direction. This is what a differential equation is. One flow corresponds to the specific value where you started your flow. The differential equation is the entire vector field.

 

[Adesh]

For what its worth, I want to quote from Higher Algebra by Hall and Knight (one of the greatest mathematical texts)

“If an equation of $n$ dimensions has more than $n$ roots it is an identity”

 

[etotheipi]

An identity is - if at all - used for certain fixed equations, or in physics when it relates different quantities. But then it's more often a law. The word identity is useless, better is formula.

Isn't there an important difference between an identity and a law? I would say that $(a + b)^2 = a^2 + 2ab + b^2$ is an identity; so for any $(a,b)$ that I choose in the plane the relationship holds.

But $c^2 = a^2 + b^2$, though a formula, would not be an identity. It would only hold for a restricted set of values $(a,b,c)$ that form a double cone in 3D space.

 

[fresh_42]

Isn't there an important difference between an identity and a law? I would say that $(a + b)^2 = a^2 + 2ab + b^2$ is an identity; so for any $(a,b)$ that I choose in the plane the relationship holds.

But $c^2 = a^2 + b^2$, though a formula, would not be an identity. It would only hold for a restricted set of values $(a,b,c)$ that form a double cone in 3D space.

It is in the end a linguistic discussion. Neither of the terms is defined, even equality is debatable. Law is it in physics, and I call $a^2+2ab+b^2=(a+b)^2$ binomial formula.

$c^2=a^2+b^2$ as you use it should correctly be written as $\{(a,b,c)\in \mathbb{R}^3\,|\,c^2=a^2+b^2\}$, which is an algebraic variety and $p(a,b,c)=c^2-a^2-b^2\in \mathbb{R}[a,b,c]$ the defining polynomial. The variety is the zero set of an ideal generated by $p$. The term equation or similar doesn't even occur in this description, except for the definition of a zero set, where it is necessary to define "vanishes at".

I said "identity" is useless, which of course is an opinion. The only situation I can think of where it may be useful is: two fourth equals one half, but two fourth and one half isn't identical. But the word equation in this context is already an exception. We write $\frac{a}{b}=\frac{c}{d}$ if $ad=bc$, but if we look at it, then $\frac{a}{b}$ and $\frac{c}{d}$ are only equivalent: they represent the same number. As soon as "represent the same number" becomes only a little less obvious, in the situation of remainders, we don't use equality anymore and write congruent: $a\equiv b \mod c\;$ iff $c\,|\,(a-b)$. This is also an equivalence relation, $a$ and $b$ are equivalent, but we don't say "equal" anymore. Only equivalent quotients of real numbers are called equal, although it is an equivalence relation, too. Some authors write e.g. $f(x)\equiv 0$ and say the function is identical zero. This is what @Adesh quoted in post #4. I admit that this makes sense: The $0$ function in a vector space of functions is the function "which is identical zero". "Which equals zero" could be misleading in this context as the connotation is, that it doesn't always equal zero, which would be wrong. However, I wouldn't call $f\equiv 0$ an identity and only use "identical to".

I stay with my opinion: identity is useless in mathematics. It is either a formula, an equation, a theorem, or equivalent. But of course, this is a debate for linguists. What's not defined doesn't count in mathematics, and we don't have a one-fits-all definition for these terms.

 

[etotheipi]

I stay with my opinion: identity is useless in mathematics. It is either a formula, an equation, a theorem, or equivalent. But of course, this is a debate for linguists. What's not defined doesn't count in mathematics, and we don't have a one-fits-all definition for these terms.

The only instance I remember of the term being useful is in the justification of comparing coefficients. If we have a relation like $Af_1(x) + Bf_2(x) = Cf_1(x) + Df_2(x)$ which holds for all $x$ (so is an "identity") and $f_1(x)$ and $f_2(x)$ are linearly independent in the function space then evidently $A=C$ and $B=D$, since $(A-C)f_1(x) + (B-D)f_2(x) = 0$ and this by definition only holds if the coefficients are zero. However if the relation is no longer constrained to hold $\forall x$ then there are no such constraints on the coefficients even if the functions are still linearly independent. In this sense, the term identity is just synonymous with the $\forall x$ part.

I think you are correct that it is linguistics. The definitions I am familiar with is that an identity is an equation that holds for all values of all variables in the equation (e.g. $\sin^2{x} + \cos^2{x} \equiv 1$), as opposed to an ordinary equation which holds for a restricted subset of the domains. Perhaps its not too useful a label, though!