Understanding differential equations backwards

[jetset]

understanding differential equations "backwards"

Ok, so D.E. all all fine and dandy, heck I've taken a whole course of D.E. in 2nd year university and even did not too badly. While I did alright, I can't say I fully enveloped the big picture of the situation at hand that comes along with D.E.s

My simplest example of a D.E. that puts things into a simple perspective is a function where 2 or more variables depend on change in reference to something else. So for instance x changes with relation to y, the result of which changes with z.

I'm looking for intuitve based explanations (that don't delve into the math to cound on "getting it", but rather words and visualizations that could further clairify things and perhaps ad some knowledge. I'm a super inntuitive and visual learner, I find that a lot of formulas make sense in their own right, but don't give me the bigger picture as often as more "philisphical" and general comments.

How can I look at a graph and say: "ah, that sir, is a differential equation for the following reason(s):..." How do I start off from not a D.E., but end up at one? How can I better intuitively understand D.E.s "backwards"?

I have the background for this stuff, and I don't exactly know what I'm missing/looking for; bear with me:P

 

[dextercioby]

I don't follow you at all.I'm sorry,but you're going to have to explain to me,better by an example,what the heck do you mean by "backwards"??From the solution to the equation??

Daniel.

 

[jetset]

I don't follow you at all.I'm sorry,but you're going to have to explain to me,better by an example,what the heck do you mean by "backwards"??From the solution to the equation??

Daniel.

exaclty.

Overall, I would like to better understand how to spot a differential equation represented by a) a graph b) good real life examples

(a lot of times the graph is the thing we do last in a DE question. Are there any graphs that are distictly and definately a DE?/ how do you know)

 

[dextercioby]

Well,u could be doing some inventing.That's what it would be.Take a function differentiate it 7 times,square the result,multiply everything with weird functions and add more terms to the equation and set everything to zero.And then try to solve the equation to find the original function.I say it's nonsense,don't u think so??

U mean draw the integral curves/surfaces??It's pretty difficult to graph some function u don't know,wouldn't u say so??

Daniel.

 

[jetset]

Well,u could be doing some inventing.That's what it would be.Take a function differentiate it 7 times,square the result,multiply everything with weird functions and add more terms to the equation and set everything to zero.And then try to solve the equation to find the original function.I say it's nonsense,don't u think so??

U mean draw the integral curves/surfaces??It's pretty difficult to graph some function u don't know,wouldn't u say so??

Daniel.

What I'm saying is are there any distinct "signiture" DE graphs?

I am saying the opposite of you last statement, I am saying looking at a surface first then saying ah yes, differential equation over here! though, i really don't need to deal with 3d, 2d is sufficient

 

[dextercioby]

Jesus,how can u say that?Take a sphere or a circle.How do you know they are integral curves/surfaces?How can u find differential equations??Taking their equation/function which defines them as geometrical locus (in some arbitrary system of coordinates) and "inventing".What's the point in that?

Daniel.

 

[arildno]

jetset:
Do you think it is worthwhile to enumerate those equations in which "2" is a solution?
That's what you're basically asking for, with a known function f playing a role analogous to the number 2.

 

[jetset]

im not wanting anything specific like "2", to be analagous, I do know the equation of a number like two is a straight, flat line. I know what an exponential equation looks like,... I'm looking for any generalities I can pick out

 

[dextercioby]

In this case,even generalities have to follow some logics.The logics says that what you're asking is a little absurd.

Daniel.

 

[matt grime]

Can i ask what you think a graph of a differential equation is? What are its axes and so on.

 

[Crosson]

Wow forumites, you are berating this poor guy for asking a fine question which you are too narrow to consider yourselves.

Consider a geometric curve for which you wanted an algebraic equation. A valid way to do this would be to geometrically construct a differential equation for which the curve is a solution, and solve the equation algebraicly.

To be explicit, consider the curve e^x. This curve can be geometrically defined as the curve for which the tangent line to each point has a slope equal to the height of the point. By rewriting this statement as a differential equation it is possible for me to obtain a power series solution that somehow (does anyone know how?) can be expressed as an irrational constant raised to a power.

Jet Set wants to go from geometry to algebra using differential equations. Obviously, this may be a difficult approach but hopefully (even in the above trivial example) we can all see that it would be a very powerful tool.

This idea is in the "thought of a thought" phase and so we should all be grateful that Jet Set gave us a real mathematical problem to solve.

 

[mathwonk]

look, suppose you have the graph of a function. then the derivatives of that function would give you a bunch of tangent lines to that rgaph. now to work backwards, suppsoe you have a family of lines in the plane. then you can ask for a graph such that those lines are tangent to it.

that is a picture or graph of a differntial equation, it is also called a vector field or direction field. so take any surface like a sphere, and imagine an arrow tangent oto each point of the sphere. then that is a differnetial equation. the solution would be a curve on the sphere such that at each of its points the vector there is the velocity vector for that curve.


this stuff is in any book on d.e.

here is a more homely illustration: imagine a road with a speed sign at every point. that is a differential equation. a solution is a car going along at exactly the right speed at every instant.

 

[cepheid]

here is a more homely illustration: imagine a road with a speed sign at every point. that is a differential equation. a solution is a car going along at exactly the right speed at every instant.

It sounds like an "imaginary solution" to me. I, myself, have not seen it happen in practice.

 

[cepheid]

How about this one: Population growth: (also in any DE book!). So you have population of bacteria, or bugs, or deer, or people. Ignoring that fancy logistic equation, just think about it this way: one organism reproduces, you end up with two or more, each capable of reproducing. If bacteria double every so often...then you get four, then eight, then sixteen...etc. Modelling this discrete situation as sort of continuous, you see that **the rate of population growth is proportional to the population itself**. I.e. in plainer language: the larger the population is, the faster it grows (because as I mentioned before, you have more reproducing organisms, and therefore a higher birth rate). Getting back to the key statement:

**the rate of population growth is proportional to the population itself**

Can you see why we call this 'exponential growth'? (everyone even knows that term in an everday sense). Well of course! any time the derivative of a function is proportional to the original function, that function is exponential...so just by inspection you can see that we have an exponential solution to the DE describing the population growth. Don't believe me? Solve it:

dP/dt = kP(t), where k is some constant of proportionality.

phew...made a non-facetious contribution to the thread. Hopefully mathwonk won't be too annoyed now.

 

[mathwonk]

geez, cepheid, you know population growth is an integer valued function, not real valued at all, hence completely inappropriate for a differential equation. maybe a difference equation.




\\JUST KIDDING!

 

[mathwonk]

now that cepheid and i have taken the question seriously, why is jetset suddenly silent?