I know nothing about differential equations

Unstable

Don’t make things to difficult. It is just a simple equation. There are of course two ways to understand Des (from the math point also from the application point of view (but this is difficult) that’s what previous writer try to explain) .

So, what is the solution of
[itex] 3\cdot b = 6 [/itex] Every kid from play school knows the answer what b is.

It is the same with Des. What is the solution of
[itex] x’(t) = x(t)[/itex] .... ??? [itex] x(t) = ? [/itex]

Is it maybe [itex] sin(t) [/itex] or [itex] exp(t) [/itex] or [itex] t^2 [/itex] ???

I know that we had math students after 3 years who could not answer this question.
They failed btw. Best answer was [itex] x(t) = \frac{1}{2} x(t) [/itex] which is definitly wrong. Why?

After understanding this simple equation you could start to understand what differential equations
are actually about. But first get familiar with the notation!

Muphrid

The problem is that we have people unfamliar with what [itex]x'(t)[/itex] means (it denotes the first derivative of the function [itex]x(t)[/itex], and is also denoted [itex]\frac{dx}{dt}[/itex], among other ways).

dawin

I think we're hinging on the equivalent of defining a word using the word in question (maddening!)

Questions such as

Makes it seem that the misunderstanding goes a bit deeper than "find a function x(t) that satisfies x'(t) = x(t)"...

Yeah, the concepts you drew a comparison between are similar, but probably still a bit too top-level at this point.

Unstable

okay
[itex] x'(t) = \frac{dx}{dt} = \frac{\partial x}{\partial t} = \frac{d}{dt} x(t) = \frac{\partial}{\partial t} x(t) = \dot{x}(t) [/itex].

Aummarizing it is the derivation of a function [itex] x [/itex] after [itex] t [/itex]

Unstable

But how could you explain people what Des are. Starting with Law of mass action? Then you
have to problem that everybody applies it but nobody really understand it what he is doing.

dawin

That's not to say you can't define them ever. But start where they're more likely to be familiar with the subject. I'm not saying what I wrote is best, either.

But if he isn't familiar with x'(t) notation, or what a function is, then giving all the different notations of a derivative might be confusing. I know it would have confused me. Then you get the "what's dx/dy? or that funny looking d (partials!) mean?" Ya dig? He or she will start getting caught up in the details rather than the concepts.

I just remember when I first started pondering DEs. And I got the formal definitions at first and they didn't help (they did, but it was harder to wrap my brain around it. I still haven't succeeded). I had to move from something specific and then say "Oh, that's like 'this' and 'that' and... Oh I see now." and then move onto the rest.

I_am_learning

Although there are already very good responses, I would like to add my take.
We solve simple algebraic equations like this
x^2+2x + 1 = 5
It means, find a number which if squared and added to its double and added 1 is equal to 5.
Differential Equations are just like that but they apply to changing quantity like distance traveled, Voltage in AC mains etc. The special feature of such things, unlike the variable x we discussed earlier is that they are changing with time and have rate of change.
For eg. rate of change of Voltage V is 5V per second, which will be written as
dv/dt = 5;

If we form equations with such quantity incorporating the rate of change in the equations, then the puzzle formed is called differential equation.
Eg. dv/dt + 2*v = 4;
The solution cant give something like V = 5 volts because the voltage is changing constantly.
So, the soultion should actually give V at various times, like
V = 5 volts at t = 0.1 seconds
V = 6 Volts at t = 0.2 seconds
V = ...
etc.
So, instead of providing solution table, we state the solution mathematically, like
V = 3*sin(t) + 2*t
Now put the required time 't' in right side and obtain V at that time.
We now call, V is a function of 't'.
The task of differential equations is to solve for the function V.

Unstable

Okay! I was not aware of that. Sorry but then forget it!

I mean the same would be: I like to play poker but I don't know what cards or money are!

jackmell

Many, many people live their lives in a cloud of ignorance about the world around them. They just do not understand. And I don't mean just math things but rather people things, social things, life things. This ignorance can cause a great deal of agony in their lives. Differential equations offer a window into that understanding and looking through it will humble you, comfort you, and give you some peace: you may not like what you see, but at least you understand why the world is the way it is.

dawin

Certainly does make the explanation difficult

Unstable

Wow...

mather

hello!

can anyone add to this excellent animation, the animated graph of the second, third, etc derivative?



thanks!

Integral

One reason we study Differential equations is that in the world around us it is easy to observe and measure changes of quantities in time or space. Changes in temperature, changes in speed, changes in concentration, etc. We have found that if we can express those changes mathematically as differentials. This leads us to an equation containing expressions of differentials of our variables of interest. If we can solve these differential equations we end up with a function of that variable in time or space.

mather

thanks!

as for my question above?

mather

anyone ???

Mark44

These deserve some comment. Assuming x is a function of a single variable t, then all of the above are different ways of writing the derivative of x with respect to t.

x'(t) and ##\dot{x}(t) ## are variations of Newton's notation. For Newton, derivatives were always time derivatives; i.e., derivatives with respect to t. Newton used the dot notation, and the "prime" notation, as in x', is very similar.

The "d/dt" notation is due to Liebniz, who developed calculus at about the same time as Newton.

The notation with the "curly" d indicates that we're dealing with a partial derivative. That is, the function being differentiated has two or more variables, and we're looking at the (partial) derivative with respect to one of those variables.

If x is a function of only one variable, say t, then the partial (or partial derivative) of x with respect to t is exactly the same as the derivative of x with respect to to. OTOH, if x happens to be a function of, say, t and v, then the ordinary derivative is not defined, but the two partials are.

In other words, this is meaningless for a function of two or more variables: $$ \frac{dx}{dt}$$
but these have meaning: $$ \frac{\partial x}{\partial t} \text{and} \frac{\partial x}{\partial v}$$
It is the derivative of x with respect to t. Derivation has a different meaning.

djpailo

OP, is this what you are looking for:
http://www.math.umn.edu/~rogness/mul...alderivs.shtml