Differential function vs differential equation

[Square1]

As the title suggests, could someone "differentiate" :) between the two phrases?

We learn about differentiation first and find out that you can get the rate of change at a certain point 'a'. Then we find out that you can obtain a function that pumps out the derivative at a defined point. What is the the difference between the differential function and, the latest topic we've started, differential equations?

Thanks

 

[chiro]

As the title suggests, could someone "differentiate" :) between the two phrases?

We learn about differentiation first and find out that you can get the rate of change at a certain point 'a'. Then we find out that you can obtain a function that pumps out the derivative at a defined point. What is the the difference between the differential function and, the latest topic we've started, differential equations?

Thanks

Hey Square1.

The big difference IMO is that one is explicit and another is implicit.

Consider the function f(x) = y = x^2 + x + 2 and the function y^2x + x^2SQRT(y+x) = y. The first is an explicit equation of y in terms of x and the second is an implicit equation of y and x in terms of each other.

In a differential equation, you usually get an implicit equation in the general case in terms of the derivatives by remembering that d/dx of dy/dx is d^2y/dx^2 and so on. It is this idea that will help you understand the extension of dy/dx = f(x) to something like d^2y/dx^2 - yxdy/dx + x^2 + 2y = 0.

If you ever do this in depth, you'll see that this implicit nature makes things really complicated especially if you are not dealing what is known as a linear differential equation.

Because real world systems are modeled often with non-linear equations, we need to know how to solve them and often this means using a computer. But the thing is we can't just plug everything into a computer: we need to know theoretically what the computer needs to calculate in order for the output on our computer to even make sense and we also need to know when the DE we have even makes sense to begin with: in other words, the DE itself might not make sense as a unique function and if this is the case then we can't even compute the function because it's not really a sound function to begin with.

 

[Square1]

Hey thanks for the reply. It took me some to let the stuff sink in/do some more problems to increase my "feel" for the matter. So your response has more meaning now :)

 

[HallsofIvy]

I can't tell you any difference because I don't think I have ever seen the phrase "differential function"! Do you mean the differential of a function or differentiable function?

Of course, the crucial part of any kind of equation is the fact that there is a "=" in it! A differential equation is an equation that includes the derivative of a function. Typically, but not always, the "problem" associated with a differential equation is to find the function whose derivative satisfies that equation.

 

[Square1]

yeah I think it's nothing more complicated than that (the definition that is!). And as chiro pointed out, they often in my classes are presented implicitly. If they were in explicit form, it would be really no additional work then just doing basic integration that we've done up till now I think.