What is the difference between a differential equation and a derivative?

[Alshia]

For example, if y=x^2, then the derivative of y is 2x. We write the derivative as either f'(x)=2x or dy/dx=2x.

Well, the differential equation is also written as dy/dx=2x. So is there a difference between a differential equation and a derivative?


!~Alshia~!

 

[Astronuc]

The derivative is just the slope of a tangent line to a function as a give point.

Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. This functional relationship is often denoted y = f(x), where f denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

. . . .

http://en.wikipedia.org/wiki/Derivative

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders, e.g., first derivative, second derivative, . . . . The challenge then is to find the function.

http://en.wikipedia.org/wiki/Differential_equation

In the case of a derivative, one knows the function, i.e., given y = f(x), y' = f'(x).

 

[Alshia]

So let's see if I'm getting this.

A derivative:

- is the slope of a line tangent to a function at a given point,
- cannot contain a third (or more) variable since it's supposed to tell us the instantaneous rate of change of a variable in relation to one other variable,
- cannot contain other levels of derivatives because of the same reason as above,
- the function which tells us about the relationship between the variables is known, and
- is a function itself.

A differential equation:

- may or may not tell us what the derivative of the function is, since it may contain different levels of derivatives and/or other variables,
- the function which relates the DV (dependent variable) to the independent variable is not known,
- an equation, not a function.


So my guess is, the reason why it's called differently from a linguistic standpoint is that a derivative is derived from a function (which of course requires the function to be known), whereas a diferential equation may not have been derived from a function (since it IS possible to measure rate of change without knowing the function).

Other than that, a differential equation MUST contain a derivative, but it may contain additional things like different levels of derivatives, other variables, etc.


Is this understanding correct?

 

[X89codered89X]

I have what may help as a perspective as I often find myself trying to explain what differential equations is to people that took calculus but stopped after that. You seem to be in that position. We'll stick with the traditional definition (something along the lines of what) Astronuc gave you: a derivative is the instantaneous slope of a function.

Let's generalize this definition a bit, and let just make clear that when i say "n-th order derivative" of y(x) , I'm referring to the 0-th order of y(x) as just y(x), i.e. $y^{(n)}(x)$, such that n=0. Notice that the equation you provided for $y^{'}(x) = 2x$ is just an equation involving one "n-th order derivative" of y(x).

A differential equation involves more than one of these n-th order derivatives. A differential equation can be expressed as a function, we'll call it g, set to 0 (if required by simply moving all terms over to one side). $g(x,y^{(n)}(x),y^{(k)}(x),...)=0$. It's still not a differential equation until I tell you that n=/=k. and keep in mind it can involve even more derivatives.

A differential equation is a generalized relationship between different orders of derivatives. "Solving the equation" means finding a closed form expression for y(x).

In most calculus problems, you're usually given f(x) or y(x) , or you've found it from the physics of a given problem, and you're asked to find the derivative - you follow rules and patterns you learn from the limit definition. Usually, however (at least in more realistic scenarios), we can't directly come up with an expression for f(x) or y(x), just the relationship between various derivatives of y(x) or f(x) and "solve the differential equation".

 

[Edgardo]

A differential equation is an equation that contains a function f(x) and one or more derivatives of f(x).

Example 1:
f(x) = -f ''(x)

This is a differential equation since it contains f(x) and the second derivative f ''(x).
The goal is to find a function f(x) that fulfills the differential equation. Such an f(x) is called the solution of the differential equation.

For the above equation a solution is given by f(x) = sin(x).
Let us verify this:
f(x) = sin(x)
f '(x) = cos(x)
f ''(x) = -sin(x)

We find that
f(x)= sin(x) = -(-sin(x)) = -f ''(x)
such that f(x) = -f ''(x).

Example 2:
Let us consider the equation
2*f(x) - x*f '(x) = 0

The equation is a differential equation since it contains the function f(x)
and its first derivative f '(x).

Verify that f(x) = x^2 is a solution for the differential equation.


Example 3:
f(x) = f '(x)

Do you know a function f(x) that fulfills this differential equation?

Example 4:
f ''(x)*f(x) - x*f '(x) = 0

Verify that f(x) = x^2 is a solution.

 

[HassanEE]

The difference is that a derivative, in an abstract sense, is a mathematical operation you apply onto a function/variable (like logarithms, exponentiation, square roots, etc). A differential equation is an equation that just-so-happens to contain derivatives. Here's a simple example:

we call x⁵ + 5x + 1 = 0 a polynomial equation because the variables are exponentiated (raised to a certain power).

we also call 5d²x/dt² - 2 dx/dt - 10 = 0 a differential equation because it contains derivatives (aka differentials). And just like any other equation, we normally want to solve for the variables being operated on.

This is a sort of heuristic explanation. I hope you find it helpful!