In the simplest qualitative terms what is a differential equation?

[find_the_fun]

I'm going to be taking a course in differential equations and I'm nervous. From previous calculus courses I know

1. the derivative is the ratio of how one quantity changes with respect to another
2. the integral is the area under the curve

So what's a differential equation? According to here "A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives. "

This doesn't make any sense because how is a differential equation different than a derivative? If you are abstractly given a function \(\displaystyle f(x)=x^2\) then you can't say it's a dirivative or antidirvate of anything.

 

[Opalg]

I'm going to be taking a course in differential equations and I'm nervous. From previous calculus courses I know

1. the derivative is the ratio of how one quantity changes with respect to another
2. the integral is the area under the curve

So what's a differential equation? According to here "A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives. "

This doesn't make any sense because how is a differential equation different than a derivative? If you are abstractly given a function \(\displaystyle f(x)=x^2\) then you can't say it's a dirivative or antidirvate of anything.

In a differential equation, the idea is to find $y$ as a function of $x$, given some information about $y$ and its derivatives. The very simplest example of a differential equation might be something like the equation $\frac{dy}{dx} = 2x$. You can easily solve that by integrating it, to find that the solution is $y=x^2$ plus a constant of integration. But suppose that the equation is slightly more complicated, for example $\frac{dy}{dx} = x +y$. How would you solve that to find $y$ as a function of $x$? A course in differential equations will show you how to do that.

[sp]In case you are wondering, the solution to that equation is $y = -x-1 + ce^x$, where $c$ is a constant.[/sp]

 

[HallsofIvy]

I'm going to be taking a course in differential equations and I'm nervous. From previous calculus courses I know

1. the derivative is the ratio of how one quantity changes with respect to another
2. the integral is the area under the curve

So what's a differential equation? According to here "A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives. "

This doesn't make any sense because how is a differential equation different than a derivative? If you are abstractly given a function \(\displaystyle f(x)=x^2\) then you can't say it's a dirivative or antidirvate of anything.

A differential equation is an equation, a derivative is NOT! Okay, having got that off my chest, I think I understand your point. "Contains derivatives", etc. is not sufficient. The crucial point is that a differential equation contains derivatives of some unknown function. Yes, "$$x^2$$" can be thought of as the derivative of $$\frac{1}{3}x^3$$ but just having $$x^2$$ in an equation does NOT make it a "differential equation". To be a differential equation, the equation must contain something like $$\frac{dy}{dx}$$, $$\frac{\partial y}{\partial t}$$, $$\frac{d^2y}{dx^2}$$, etc., where y is the "unknown" function.

More generally, a "functional equation" is an equation that gives us some information about a function. "f(x+ 1)= f(x)" is a functional equation. $$\frac{d^2y}{dx^2}+ 2\frac{dy}{dx}+ y= x^2$$ is a special kind of a functional equation (since it gives information about the function y) called a "differential equation" because that information is actually about the derivatives of the function y.

 

[find_the_fun]

Glad I asked. I had the first lecture today and the prof skipped over the definition of a differential equation (in fairness it was a substitute prof).

What is an ordinary differential equation? Does all that mean is it doesn't have partial derivatives?

 

[Opalg]

What is an ordinary differential equation? Does all that mean is it doesn't have partial derivatives?

Yes. You will often see the abbreviations ODE and PDE for ordinary/partial differential equation.

 

[find_the_fun]

I'm writing myself notes now and here's the first:

Differential equation: a functional equation that relates an unknown function to its derivatives

 

[zzephod]

An ordinary differential equation is an equation of the form (or that can be rewritten in the form):

\(\displaystyle \large f(x,y,y',...,y^{(n)})=0\)

A partial differential equation is similar but with partial derivatives with repect to the variables appearing (including mixed partials).

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