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In the simplest qualitative terms what is a differential equation?

find_the_fun

Active member
Feb 1, 2012
166
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
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.

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

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
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, "[tex]x^2[/tex]" can be thought of as the derivative of [tex]\frac{1}{3}x^3[/tex] but just having [tex]x^2[/tex] in an equation does NOT make it a "differential equation". To be a differential equation, the equation must contain something like [tex]\frac{dy}{dx}[/tex], [tex]\frac{\partial y}{\partial t}[/tex], [tex]\frac{d^2y}{dx^2}[/tex], 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. [tex]\frac{d^2y}{dx^2}+ 2\frac{dy}{dx}+ y= x^2[/tex] 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

Active member
Feb 1, 2012
166
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
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

Active member
Feb 1, 2012
166
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

Well-known member
Feb 3, 2013
134
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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