Is the Derivative of a Function a Differential Equation?

[rakeru]

Is the derivative of a function a differential equation? I guess it would be because it involves a derivative, right? Would the solution to the equation just be the original function? Is solving a differential equation just another way of integrating?
Like with finding solutions of separable ones.. it's just integrating both sides. And with finding other solutions with exact and linear equations, there is always integration.

 

[rakeru]

I am talking about first order ones.. I don't know how to solve second order ones.

 

[pasmith]

The derivative of a function is itself a function: $$f' : x \mapsto \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

A differential equation is an equation in which derivatives of a function appear.

Thus, given $g$, $f'(x) = g'(x)$ is a differential equation with solution $f(x) = g(x) + C$.

 

[HallsofIvy]

What a strange question! No a "derivative of a function" is NOT a "differential equation because a function is not an equation!

 

[LCKurtz]

Is the derivative of a function a differential equation? I guess it would be because it involves a derivative, right? Would the solution to the equation just be the original function? Is solving a differential equation just another way of integrating?
Like with finding solutions of separable ones.. it's just integrating both sides. And with finding other solutions with exact and linear equations, there is always integration.

If you have $y' =f(x)$ then $y = F(x)+C$ where $F$ is an antiderivative of $f$ and $C$ is an arbitrary constant. So you get the original function by antidifferentiating (integrating) to within a constant. For a simple DE in that form, you do solve it by integrating, but I wouldn't say it is just another way of integrating. You use integrating to solve it. But differential equations can be more general so that you can't solve them in practice by integrating. What I mean by that is, for example, there is no general method to express the solution of a general DE like $y'=f(x,y)$ with integrals. Even so, I would agree with the statement that in some sense, "there is always integration" underlying the problem. Not sure how meaningful a vague statement like that is though.

 

[Simon Bridge]

Is the derivative of a function a differential equation?

No.
A differential equation must explicitly include the derivative.
i.e. the derivative of the function x² is 2x ... you will see that 2x does not include a derivative.
But y'=2x is a differential equation. The LHS is just some notation that tells you that the RHS is the derivative of y.

I guess it would be because it involves a derivative, right?

To be a differential equation it has to include the dervative and be an equation.
the derivative (wrt x) of y²(x) is 2y.y' explicitly includes a derivative but it is not an equation.

Would the solution to the equation just be the original function?

Lets see - the derivative (wrt x) of y²(x)=x would be 2yy'=1 ... this second is a differential equation.
What is it's solution?

Is solving a differential equation just another way of integrating?

Solving a DE means that you correctly figure out what expression makes the equation true.
This could amount to integrating - but need not involve the formal process of solving an integration.
You'll see what I mean as you move on to more complicated DEs.

 

[homeomorphic]

I think you mean to ask if f'(x) = g(x) is a differential equation. The answer to that question is yes. The expression f'(x) isn't even an equation, let alone a differential equation.

As far as solving differential equations goes, there isn't always integration. A lot of them are solved by just guessing the form of the solution and then getting the constants so that it satisfies the initial conditions.