The practical use of Differential Equations?

[Razor]

Done a search on Google and most of what i can come up with deals with simulations. My friend needs it for his school work, he says he needs to find examples of where differential equations are used for practical applications, calculate the equations, provide examples, and why differential equations are used?

Can any physcis, maths, computer, electrical experts help.

He says that the example he was given was the loading of a capacitor.

 

[marksmanHL2 :)]

Errrr, cant they be used to find the maximum and minimum solutions to problems?


Maybe I should shut up... I'm crap at maths.

 

[KagePrototype]

Differ-wha?

I lost track around Year 8. :|

 

[the_lone_wolf]

The Lagrangian equation can be used to derive shroedinger's equation (fairly fundamental of quantum mechanics) another way

maxwells equations are easier to work with in differential form

not sure about any others, these are off the top of my head

 

[baron insig]

Im doing Quantum mechanics in Physics right now it rather interesting but not very " wowzer" yet

 

[Pi Mu Rho]

When bored one afternoon, myself and a friend did some calculations (using differential and integral calculus, of course) to determine how high a tennis ball would bounce back up when dropped from the window of her kitchen on the 14th floor.
The practical tests were far more amusing...(and showed that our calculations were roughly correct, too)

 

[Dan]

simulations are pretty practical if you are planning on recreating that simulation in real life with a 12 million dollar budget or something.

 

[SidewinderX]

ok, here we go.

http://www.dpetkofsky.com/scans

not sure if this is *exactly* what you're looking for...

Scans 000-005 are from the college book about spring/mechanical systems

Scans 006-008 are from my book from last year... mainly basic growth and decay problems/solutions.


If these aren't what you're looking for, just say, and i'll try and find more.

 

[epsilon]

Hi all,

I'm Razor's friend with the assignment.
I'd like to thank all of you for helping out with this. Please keep the ideas and solutions coming :thumbs:

Greetings,

Epsilon

 

[The Mullinator]

Hi all,

I'm Razor's friend with the assignment.
I'd like to thank all of you for helping out with this. Please keep the ideas and solutions coming :thumbs:

Greetings,

Epsilon

You have the same assignment even though your location says Belgium and his says the UK?

 

[marksmanHL2 :)]

I was wondering that...

 

[Razor]

You have the same assignment even though your location says Belgium and his says the UK?

Well spotted genious, the jig is up, i knew i couldn't put one past you :rolling: .

No, i don't have the assignment, he asked me for help on it over msn as i am super smart, but not smart enough though for this question , so i asked you guys who are super, super smart.

 

[kirovman]

Differential equations are used in almost everything.

A very common and simple application is velocity and accelerations.

Velocity is the first differential of displacement w.r.t. time.
Acceleration is the second differential of displacement w.r.t. time.

Other applications:

Heat Capacity = d (Internal Energy)/d (Temperature)

Schrodinger's Equation as mentioned before, to obtain quantum mechanical energy eigenstates.

There are countless more applications, such as nuclear scattering, where you obtain a differential scattering crioss-section.
Or more simply, a rate of population change (eg in Half-Life decay of a radioactive sample)

Graphically speaking, differentials are gradients of one variable with respect to another.

If you differentiate with respect to time you obtain a rate of change.

 

[Dinkleberry]

The Lagrangian equation can be used to derive shroedinger's equation (fairly fundamental of quantum mechanics) another way

maxwells equations are easier to work with in differential form

not sure about any others, these are off the top of my head

SKOO!

How about in understandable terms for someone who isn't doing physics at university?

Here's some differential equations that I have been learning about recently in college:
Newton's law of Cooling.
Population Growth Models.
Radioactivity (Half Lifes in fact can be found from them no pun intended)

Also, in musical terms, differential equations are used in modelling the sound of real instruments.

 

[epsilon]

ok, here we go.

http://www.dpetkofsky.com/scans

not sure if this is *exactly* what you're looking for...

Scans 000-005 are from the college book about spring/mechanical systems

Scans 006-008 are from my book from last year... mainly basic growth and decay problems/solutions.


If these aren't what you're looking for, just say, and i'll try and find more.

Thanks for the scans! I'm going through them right now.
If you have more, they're always welcome.

 

[Seeky]

Differential Equations are used wherever there are multiple rates of change in a given environment.

A common example, are chemical plants having to monitor the concentration and/or temperature of certain vats, as additional are added and stirred into the tank.

This is complicated, temperature that a vat might rise whenever a certain chemical is added into it - however, the increase is less significant when there is a greater volume of liquid in the tank. But, as soon as you add chemicals into the vat, you are simultaneously changing the volume, so trying to calculate how much the temperature changes over a given time is not a simple linear equation. You need differential equations to model how the temperature changes with concentration, how the concentration changes with volume..and then how the volume reduces temperature. It's one great big circle.

Another good example, is identifying stress/fracture points in various forms of architecture and engineering. If you place a heavy object on a bridge, you'll create stress in various directions at that point on the bridge. In a simplified explanation, the stress propagates itself throughout the entire bridge, cause other points to undergo stress. These new points will create stress that affects every other point on the bridge, including the original point. You need differential equations to model how one change affects another.

Finally, in electrical engineering, you'll find that circuits frequently have components that depend on one another. An inductor on a circuit will act in different ways depending on how a capacitor many points away is acting. Likewise, the capacitor will accept varying amounts of charge, dependent on what the inductor is doing to the rest of the circuit. You need differential equations to model how two things that are changing simultaneously, affect each other.

There are far more applications than these, and many of them utilize computer software to create and solve these equations.

 

[ZoFreX]

Someone in my class got pissed off with all the "maths-land" work we were doing, ie it had no use in the real world, so the teacher found an example of using complex numbers (aka imaginary numbers) for a real problem, and it also gives an example for your problem.

2nd order differential equations with no real roots are used to model damped harmonic motion - e.g. any real-life oscillation. Used for pretty much everything in engineering I guess.

a * d2y/dx2 + b * dy/dx + c * y = f(x)

Roots are not real when b^2 - 4 * a * c is less than zero. Don't ask why, I missed the lesson where he explained it to us.