# How do the d terms in differential equations work?

1. Jul 25, 2011

### Chrisistaken

How do the "d" terms in differential equations work?

Hi,

I was hoping someone could explain how the "d" terms in differential equations work? For example,

d2y/dx2 = 4x3 +1

To solve I have been rearranging to get,

d2y = (4x3 +1)dx2

and then doing a double integral of each side.

I sort of assume that the double integral of d2y is equal to y rather than y2/2 but I don't really understand why this is the case.

Can someone please explain this to me and also why the powers are applied differently to the numerator and the denominator?

Regards,

Chris

2. Jul 25, 2011

### charlesmanima

Re: How do the "d" terms in differential equations work?

The integral of d2y is dy. Once again integrating you will get y. The way you are rewriting the equations can make confusion. To avoid this confusion rewrite d2x/dx2 as d/dx(dy/dx). Now you can integrate two times with respect to x.

3. Jul 25, 2011

### chiro

Re: How do the "d" terms in differential equations work?

Umm isn't d/dx(dy/dx) = d^2(y)/dx^2?

4. Jul 25, 2011

### Hurkyl

Staff Emeritus
Re: How do the "d" terms in differential equations work?

You're grouping symbols wrong. Roughly speaking everything in
$$\frac{d}{dx}$$​
goes together into one symbol. (though, I suppose, x is a slot for inserting an appropriate variable symbol)

AFAIK, the assortment of coincidences that allow us, when there is only one independent variable, to get sensible results by viewing $dy/dx$ as if it was the quotient of $dy$ and $dx$ don't work very well when applied to second derivatives.

You're better off working one derivative at a time -- e.g. worry first about
$$d \left( \frac{dy}{dx} \right) = (4x^3 + 1) dx$$​

It might help to introduce a new variable
$$z := \frac{dy}{dx}$$​
so the previous equation becomes
$$dz = (4x^3 + 1) dx$$​

5. Jul 25, 2011

### HallsofIvy

Staff Emeritus
Re: How do the "d" terms in differential equations work?

No, this is not valid. The first derivative is defined as a limit of a fraction and so can be "treated" like a fraction (to show that any property of a fraction is true for a derivative, go back before the limit, use that property of the difference quotient, then take the limit). The differentials, dy and dx, are defined from the derivative to make use of that.

But higher order derivatives are not defined like that and cannot be treated as fractions. In particular, saying that $d^2y/dx^2= f(x,y)$ does NOT lead to "$d^2y= f(x,y)dx^2$.

No, thats not the way double integrals work. You have to have two different variables, not $\int d^2y$

In order to remind you that higher order derivatives are NOT just fractions!

6. Jul 25, 2011

### Chrisistaken

Re: How do the "d" terms in differential equations work?

Thanks all for the quick replies. I think Charlesmanima pretty much summed up what I was after.

Just to clarify though, are the d/dx, dx and dy terms, just symbols that cannot be manipulated by the usual mathematical methods (excluding convenient coincidences)?

Regards

Chris.

7. Jul 25, 2011

### pmsrw3

Re: How do the "d" terms in differential equations work?

Also (or maybe as a specific example of what HallsofIvy is saying), it makes the units work out right. For instance, $\frac{dx}{dt}$ is velocity, which has units of distance/time, like $\frac{x}{t}$. $\frac{d^2x}{dt^2}$ is acceleration, which has units of distance/time2, like $\frac{x}{t^2}$.

8. Jul 25, 2011

### Chrisistaken

Re: How do the "d" terms in differential equations work?

So for a fail safe method of working with derivatives, you would need to go back to the limit?