Beginner question on differential equations: they look useful to search for a function for solution of an equation, but why are derivatives mandatory?

Question

At 54 yo, differential equations are still a wall for me, I have difficulties to go through.

One video for beginner helped me, explaining me that

in common equations, we are searching for the values of a variable $x$, for solutions
in differential equations, for the values of a function $f(x)$

But the video was also telling and showing that :

a $f(x)$ solution searched for, could also be named $y$, as $y = f(x)$
samples of resolution of a differential equation, where one of the member had a derivative of $f(x)$
in example : $y′+2y=x^2−2x+3$

My question might look strange, I try to ask it the clearest manner I can:

$y = f(x)$, $y' = f'(x)$, ...

Must a differential equation carry the function itself and some derivatives (of any order) of itself?
Said another manner:
is $h(x) = \frac{f(x)}{g(x)}$
a differential function too?

If I say: I'm searching for $h(x)$ for solution of a function that solves the equation above?
Said another manner:

Why searching for functions as solutions
does require to interact with their derivatives?

Is it because differential equations are not [only or just] for searching functions for solutions of anything that is an equation,
but have a special purpose I don't see?

You could imagine equations where the unknown is a function and the equation doesn’t involve derivatives. For example, we could try to find a function $f$ such that $e^{f(x)} = x^2$ for all $x \in \mathbb R$. This is not a differential equation. Differential equations are an especially interesting type of equation because: 1) They are very useful for modeling the physical world; 2) it turns out that differential equations is a mathematically rich subject and there are many interesting and useful theorems that can be discovered and proved. — littleO, Mar 18 '24 at 07:46
"A differential equation is an equation relating some function f to one or more of its derivatives." Krantz - Differential equations demystified (however, be aware of this "counter example": https://math.stackexchange.com/q/2434615/532409) — Quillo, Mar 18 '24 at 07:50

score 4 · Accepted Answer · answered Mar 18 '24 at 08:45

One video for beginner helped me, explaining me that

in common equations, we are searching for the values of a variable $x$, for solutions

in differential equations, for the values of a function $f(x)$

That statement is true, but misleading. Here is a more accurate statement:

An "ordinary" equation has variables which stand for numbers, and a solution to the equation is a number (or collection of numbers) which makes the equation true.
A functional equation has variables which stand for functions, and a solution to the equation is a function (or collection of functions) which makes the equation true.
A differential equation is a type of functional equation. Specifically, a differential equation is a functional equation that involves a derivative.

To answer your specific questions:

Must a differential equation carry the function itself and some derivatives (of any order) of itself?

To be precise, a differential equation must contain a derivative somewhere, because that's what the phrase "differential equation" means. It doesn't have to contain the original function itself. For example, the equation $f''(x) = f'(x)$ is a differential equation.

Said another manner: is $h(x) = \frac{f(x)}{g(x)}$ a differential [equation] too?

No, that equation is not a differential equation, but it is a functional equation.

(You wrote "differential function," but I think that was a typo for "differential equation." Please correct me if I'm wrong.)

Why searching for functions as solutions does require to interact with their derivatives?

Is it because differential equations are not [only or just] for searching functions for solutions of anything that is an equation, but have a special purpose I don't see?

Searching for functions as solutions certainly doesn't require us to use derivatives at all. Derivatives aren't mandatory. It's just that if an equation involves a derivative, we call it a "differential equation," and if it doesn't involve a derivative, we call it something else.

Differential equations show up all the time in physics. For one reason or another, nature is full of functions which have particular relationships to their own derivatives.

Non-differential functional equations are important, too. For example, exponential functions satisfy the very important functional equation $f(x + y) = f(x) f(y)$. The gamma function is the most notable solution to the functional equation $\Gamma(z + 1) = z \Gamma(z)$.

Great answer! Thanks a lot! – Marc Le Bihan Mar 18 '24 at 09:55 — Marc Le Bihan, Mar 18 '24 at 09:55

Blitzer · Answer 2 · 2024-03-18T09:19:21.877

Q1 / Q2: To be a differential equation it must have a derivative of a function in it.

So $y''+y=x^2$ is a differential equation but $h(x)=f(x) g(x)$ is not.

I'm not sure I understand Q3 but perhaps you are asking why differential equations are important. It just turns out that they are very good models of the world, particularly in physics but also in chemistry, economics and so on. You probably know Newton's second law which can be written $F(t)=m x''(t)$ for example. Physicists want to be able to solve differential equations because they describe the physical world in exquisite detail.

Beginner question on differential equations: they look useful to search for a function for solution of an equation, but why are derivatives mandatory?

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