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Ace17

What the hell is a differential equation???

10 posts in this topic

A differential equation is a function that expresses the relationship between a function and the rate of change of that function such as acceleration, velocity and position of an object. If you have a function that defines the relationship between the velocity and position of an object then solving that equation means finding a function that expresses the position as a function of time. As a simple example if v=t*s where v is the velocity, t is time and s is position then a solution is s=C*e^((t^2)/2), i.e. s''=t*C*e^((t^2)/2) which says v=t*s.
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In short, a differential equation is the gradent function of the differentiated equation. With respect to acceleration and velocity, the gradient of a velocity/time graph is the acceleration, so if you differentiate the function of the v/t graph, you get an equation for the acceleration.
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An occurrence to drink a lot of beer over.
(second year university mathematics, if you're getting a math degree)


dy/dx = x

is a differential equation (a very simple one). [the d means differential, and more-or-less means 'the change in...']


To solve it you bring the dx over to the over side, just like in algebra.

dy = x * dx


Then you integrate it (that funky S looking symbol)

S dy = S x * dx ->
y = 1/2 x2


That's a trivial ordinary differential equation that's covered in Calc 2.

Usually people are referring to non-trivial ordinary differential equations, or partial differential equation (which are useful when you have multiple degrees of freedom, for instance collision detection)

And a gradient has a very specific definition!

∆f = <@f/@x, @f/@y, @f/@z>

is the gradient! [of f(x,y,z)]

I've been told the US & Britian universities interchange gradient and differential - apparently to maximize confusion...

Magmai Kai Holmlor
- Not For Rent

[edit] The formatting is as close as I can get it - what do you think it would take to get Myoptic to switch the board over to unicode?

Edited by - Magmai Kai Holmlor on October 1, 2001 6:35:35 PM
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While the answers above do convey ''some'' of the answer to the question, I do not believe they convey it very clearly, or thoroughly enough to provide an understanding of what a differential equation is. Magmai''s answer also suffered from the problem of notation that didn''t format well.

A differential equation expresses the rate of change of one quantity with respect to infinitesimal changes in another quantity.

When many people think of differential equations, they are thinking of Ordinary Differential Equations. If for some independant variable, t, there exists a (possibly vector) dependent variable x, which is to be a function of t so that x = x(t), then under certain conditions we may be able to write a function representing the change of x with respect to infinitesimal changes in t (the derivative of x with respect to t) and this function often has the notation:

dx/dt = f(x,t)

This is an example of a differential equation.

On the other hand, if there exists a function z = f(x,y), for independent variables x and y and dependent variable z, then we refer to rates of change of z (derivatives) with respect to x and/or y as partial derivatives. Most commonly we determine partial derivatives with respect to one independent variable by holding all other independent variables constant, however this is not strictly necessary. We can infact determine the rate of change of the dependent variable with respect to any direction on the independent manifold (which means in lay terms we can find the gradient (at a point) in any direction on the surface z = f(x,y) ).

(I cannot write a partial differential equation as I cannot generate the correct lowercase Greek delta symbol, sorry!)

Most importantly, one should consider ''why'' we use differential equations. Ordinary differential equations and differential calculus can be used to represent and reason about change in our world over time. In particular, we formulate models of dynamic systems so that we can predict their behaviour under certain conditions and at certain times. Differential equations are a representation of these models and differential calculus allows us to perform the reasoning about the system.

Partial differential equations are often used when we want to analyse and understand the internal behaviour of a system; why one quantity varies as other quantities vary or are held constant.

If you have any specific questions about differential equations or their use I would be happy to anser them.

Regards,

Timkin
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quote:
Original post by Magmai Kai Holmlor

I've been told the US & Britian universities interchange gradient and differential - apparently to maximize confusion...




Erm, I've was tought that the derivitive of an equation is the gradient function of that equation. So, a very simple example:

f(x) = 2x3+3x2-6x + 12

f'(x) = 6x2+6x-6

Plugging values of x into f' would produce the gradient of f at that point.

Differentiation is used to find tangent lines to curves aswell, because it finds the gradient, otherwise the Newton-Raphson process of finding roots wouldn't work. Anyway, I'm probably just wrong

Phil.



Edited by - python_regious on October 1, 2001 1:46:48 PM
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That's exactly right, just by convention we (in the US) don't call that the gradient (even though it is a single dimension gradient) - we explicitly call it the derivative and reserve gradient for multiple dimensions only.
I had a British numerical analysis book in school, and the author kept saying 'gradient' and I couldn't figure out where the other dimensions were

Magmai Kai Holmlor
- Not For Rent

And a lower-case delta doesn't do you any good, those differntial, and partial differential symbols aren't greek letters - unless there's a different convention in Australia.
A 'd' is close enough for a diff. and the '@' is as close as you get in ACSII to a partial (rotate it 180o and it's pretty darn close.)

Edited by - Magmai Kai Holmlor on October 1, 2001 6:40:23 PM
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