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Ordinary Differential Equations – A Review of Basic Solutions

Posted September 01, 2010 12:01 AM by mcgratp45

Background

One of the most straightforward means of describing the behavior of a system is the differential equation. Applications of the differential equation can extend to different fields that are not limited to physics and engineering. Many facets of biology, sociology and economics can be described using the differential equation (e.g. carbon dating, population growth, compound interest on a financial loan). This blog entry will focus primarily on second order differential equation review with a mini-recap of first order equation theory.

First Order Differential Equation Review

The general solution of a first order differential equation can be demonstrated through a simple example using the variables y and t. There exists for a function y = F(t) a solution which satisfies the condition:

The solution can be attained by simply isolating the differentials and integrating both sides, as follows:

The above function must satisfy the condition of equivalence to its own derivative. A common elementary solution to the differential equation is:

in which C is a constant with the value . If there are initial conditions given, plugging them in the general solution for y and t yields a solution to the more specific case under observation.

Second Order Differential Equations

Second order differential equations are applicable throughout engineering and science and are used in applications where there is either electrical or mechanical oscillatory motion to be modeled. A second order differential equation takes the general form:

This is called the characteristic equation of a second order differential equation. In order to solve equations of this nature, one can split out the differential terms into variables and solve for them in the same manner they would the roots of a polynomial equation:

In a second order differential equation, there are three different cases of roots that can occur, and the root types are conditional based upon the square root value in the numerator. These cases represent something called damping, a phenomenon which reduces the amplitude of waveforms in a system. The pink sinusoidal waveform does not appear to taper off in the image below. The other three waveforms are relevant to the cases that will be discussed in this entry.

Case #1: Real and Distinct Roots

The first case is where both roots to the polynomial are real and distinct, and results when:

Since the value here is positive, this guarantees that no complex numbers will appear in the solution. This case is called the over-damped case.

Constructive Over-damping

Over-damping can be a positive in certain cases. One example would be the need to completely avoid overshooting of the steady state value within the system. An over-damped response will slowly reach its steady state in an exponentially decaying fashion, thus avoiding problems associated with excessive motion or current flow, especially in systems that are sensitive to sudden and rapid change. Temperature is an example: some things such as pressure cookers and kitchenware need to slowly be brought up to the correct temperature lest the material will fracture under excessive initial heat.

Destructive Over-damping

Over-damping can be destructive in that it takes excessive amounts of time for the system to reach its steady state, especially in cases where transient system effects should dissipate much sooner. Though there are not generally oscillations associated with over-damped motion, the strain associated with it can be stressful on system equipment if sudden surges or forces take place. Over-damped automotive systems tend to be excessively stiff and resistant to bumps in the road, thus causing an extremely rough ride.

Case #2: Repeated Roots

The second case is where instead of two real and distinct roots, there is a single real repeated root. This case results when:

This case is called the critically damped case.

We have found a single solution for y(t). However, according to the general solution of a differential equation, there should be another solution and it should be linearly independent of the above y(t). The next choice for a solution would be a solution of the form:

This makes sense as the function is assumed to be the product of y and some unknown t. One can prove the above solution by taking the derivative of and substituting it into the characteristic equation for a second order differential equation. This leads us to our final form of the critically damped case, which is:

Constructive Critical Damping

Critical damping is usually the ideal choice when designing a system and this can be seen especially in the automotive department with shock absorption. In an over-damped system, a small bump in the road would have a large effect on the system. The critically damped case ensures the fastest return of the system to the steady state along with eliminating any oscillatory motion.

Destructive Critical Damping

While critical damping often yields the best performance in systems, sometimes it is not the preferred method for the case under observation. In systems where electromagnetic interference poses a threat, the over-damped case can be preferred to the critically damped case. This can help cushion the transition between signals and kill some of the high frequency material. In excess, this degrades the system performance.

Case #3: Complex Roots

The final case under discussion is where the roots to the polynomial are complex numbers. This results when:

Taking the square root of a negative number often yields a solution with both a real and imaginary component (e.g. a + bj and a - bj). This case is called the under-damped case.

Using Euler's Formula and the form a + bj and a-bj from above, a new variation of the second order solution emerges for complex numbers:

Constructive Under-damping

Under-damping can be useful in some applications as opposed to critical damping. If the input to a system changes rapidly, selecting a slightly under damped response would be wise as it helps gain the fastest initial response. If initial speed is the most important part of the design, the under damped case is ideal.

Destructive Under-damping

While under-damping yields the quickest initial results in a system, the loss of functionality associated with overshooting the ideal output is a large trade off. For example: in an electric circuit, under-damping of the power supply may cause the output is allowed to swing past the final allowed voltage. In this case, damage to the equipment could occur. Significant amounts of under-damping in an automobile will cause the vehicle to bounce repeatedly after hitting one small bump in the road.

References:

http://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx

http://www.sosmath.com/diffeq/second/second.html

http://en.wikipedia.org

Images:

http://www.gender.org.uk/about/06encrn/61_cntrl.htm

http://en.wikipedia.org/wiki/Damping

http://homepages.which.net/~paul.hills/Circuits/PowerServo/PowerServo.html

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#1

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 4:40 AM

Please can I be excused 2nd order please?
I have a note from my Mum saying my furry brain hurts.
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#2
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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 9:34 AM

I know how you feel. Choosing this as a topic forced me to really re-learn this stuff :)

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 12:55 PM

The first year after I got my MS degree I taught a course in introductory college level Physics. I felt I learned more about Physics that year than in the previous 6 years.

I suppose that's why in the medical schools they follow the method of 'See one, do one, teach one'.

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#12
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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 1:46 AM

Visualizing & performing this gave me a headache. So I took a few aspirin ... made my stomach so upset that I had to regurgitate. I feel better now. Seedoteach-Doctor humor.

Did the same thing ... teaching Physics ... bright students tend to have that effect. Good to see you still looking to learn new things (as evidenced by your attention to this article).

BTW, I now recommend niacin and H2O for headaches, then you can go chew on the willow bark with your eyes closed. =;o)

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#4

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 2:53 PM

Good review, I liked the auto examples : blown shock absorbers = under damped response.

One sentence has me scratching my head : "This can help cushion the transition between signals and kill some of the high frequency material."

Can you elaborate?

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 3:23 PM

Certainly.

My wording of that sentence was a bit vague - I was talking about control systems and the like. It doesn't really 'kill' the high frequencies so much as the damaging EMI they create. Since over-damped systems tend to be stiffer, they are useful in cases where lots of damage could occur and more control is necessary. The value is slowly brought to it's final state, and in most other cases undesirable as it causes a response that is unnecessarily sluggish. If a system is too heavily damped, however, especially a sound system, the sound begins to deteriorate.

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 12:07 AM

"Since over-damped systems tend to be stiffer, they are useful in cases where lots of damage could occur and more control is necessary."

I was thinking the same thing looking in my rear view mirror in my BMW doing 60mph on a 30mph entrance ramp curve at the poor bastard in the Camaro who was so smug when he passed me on a straight-away but now has to figure out how to get out of his car now that it has become one with the guardrail. True story, long time ago.

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#6

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 8:57 PM

The road to insanity is potholed with IDEs (improvised differential equations).

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 11:47 PM

Yes. Yes it is.

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#7

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 9:05 PM

Well, some of this Mcgratp45 reminds me of a guy I knew who had a bunch of graphs and equation like writings about motion pictures. I shall withhold the story in detail, but only say things did not particularly turn out well for him.

I can't really read equations myself and typically glaze over in attempts to do so. Still I attempt to understand what you are saying of value. I simply ended up in my entertainment read thinking of why shock absorbers on cars are a hydrolic and spring thing.

I read Absolam Absolam by William Faulkner the same way I read your post. I only really understood that on the last page.

P.S. Never read you before but suggest around here you correspond with Jorrie, or Roger Pink. They can read your equations.

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/01/2010 11:42 PM

Whatever you take away from it is fine, whether it be entertainment or further knowledge of math-like things. I'm here to entertain and inform

My blog entries won't all be this math intensive, just figured I'd start with something almost completely objective. You know, ease my way into the world of blogging carefully. I've read Roger and Jorrie, they're very involved in physics and make some good conceptual connections. I hope to get there someday too!!

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#22
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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 8:05 PM

What's up with the fear of blogging? I was never afraid to write a letter. I wrote for a blog for a year and a half starting three days after I actually had access to an internet connected computer. Then as a wedding present I was given a website. My website is a databased driven hybrid site these days. I suppose I blog blog here on CR4, though I did make a little blog that I forget the password too.

I feel vindicated that I am dependent on word explanations noting a mistake found in the equations. Never knew algebra or vectors hardly from the get go, so little to forget.

I did take Microbiology as an Elective, so I have a least a foot to stand on when venturing into areas where I had no base of education or experience.

Don't really care if you write math intensive or not. I'm myself around here, and you can be too.

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#11
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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 1:09 AM

" suggest around here you correspond with Jorrie, or Roger Pink. They can read your equations" That's true of course, but implies that most others wont be able to read it.

This sort of maths is typical 1st year stuff for all uni level engineers (especially the electrical variety) so it's common knowledge and commonly understood (even if not always fully remembered by this old bloke).

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#13

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 2:47 AM

Nice post. Cleared a cobweb or two.

Regarding control dampening: I think the secret is (if it is still a secret) combined dampening. The first thought I had was similar to traction control systems where the user chooses based on current conditions - using varying amounts of over dampening, constructive under dampening and critical dampening in parallel electronically, but in series mechanically via progressive spring 'rates'.

I think any design consideration is going to have to consider the user or autonomy first, before you can say which is better.

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#14

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 9:33 AM

mcgratp45

mcgratp45,

Thanks for starting this blog. I am trying to get my mathematical skills back to where they were once upon a time!

I went back to an AS course a year or so back, passed that 1 point short of a distinction; went to move on to an A2 course - but no takers so I will have to do this through books and take an exam later.

Your 2nd order differential equations will help move me on to the next stage.

Can I stay in touch please?

Sleepy

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 12:50 PM

Sure! I'll be posting things like this every so often. I'm glad this helped you out - people who need review like myself that majored in engineering or mathematics were essentially my target audience for this specific blog entry.

Feel free to follow along, I'll probably post every week and a half or so. If there's a topic you don't see on CR4 already related to math (Roger and Jorrie cover a lot in their blogs) throw me a private message and I'll see what I can do for ya as long as it isn't doing your homework assignments for you! :)

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 1:16 PM

mcgratp45:

Hi and thanks

Any homework will be self generated!!

Talk to you later

Tell me how do you generate the Mathematical symbols and equations, I have a very old copy of MathCad wich i keep meaning to start using, but I notice that your text works fine in Word, i have not tried changing, editing it.

Sleepy

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#16

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 1:11 PM

Just one question on notation.

In your form for the 2nd order diff eq,

should the second expression be:

ay" + a1y' + a2y = G(t) ?

I'm thinking that you left out the variable y that goes with the coefficient a2.

Thanks.

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 1:13 PM

It should. Had to use MS paint to do the equations, lost sight of what really mattered. Changing it now!

Thanks.

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#19

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 2:23 PM

It's a little comforting to learn that I'm not the only experienced engineer who's forgotten more about differential math than I remember. Starting a master's degree (in engineering), I know I'm going to have to re-learn all this stuff.

For me, your review needed to start a year earlier.

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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 3:53 PM

Lynn,

join the club that has forgotten more than it has remembered!!

I will need to update my books, esp as the last Maths course was a pure Maths plus Mechanical and I think that I need to do a Pure Maths course!

Good Luck to all of us.

Sleepy

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#21

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 7:45 PM

I just knew there would be solution one day

Not that it helps much

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#23

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/02/2010 8:31 PM

In grade school they told me that if I want people to understand me just remember numbers are for counting and letters are for writing.

30+ years later I am still following that rule and I find it works very well!

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#24

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/05/2010 4:18 PM

Hi mcgratp45,

Just wanted to say thanks for the discussion about over-damping, under-damping and critical damping, which gives me an inkling of why differential equations have meaning for engineers and are actually useful.

My contact with calculus was many years ago as a teen, a required course that I could not for the life of me figure out any reason why I would force myself to learn enough to pass, except that not doing it would bar me from other courses. It marked the final breakup in my chequered love affair with math. In all other contexts that I've had to use a differential, I handle them as I would if forced to interact with a cranky and venemous snake: keep confined in box, follow strict handling procedure, never allow it to roam freely among the other thoughts.

So it's nice to read your words and find it so reasonable. Good start to your blog.

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#25

Re: Ordinary Differential Equations – A Review of Basic Solutions

09/05/2010 6:40 PM

I have been following your blog because I am compelled to make sense out of some of the studies I labored over at the University of Massachusetts so many years ago it seems like another life.

In a sense I am glad some of that didn't change, as so much has!

But looking at challenges I have faced in the not too distant past, they include bouts with control theory – which I can honestly say I finished the course becoming more confused than when I started.

Now, for myself (as the review and writing will do me good), I will outline PID control, and ask how much math does one have to cover before you can get into control theory? I know the math in that controls course was, shall we say, dizzying.

But the term 'dampening', and the result, is also used in PID control.

The question is whether or not the coefficients are alike.

To be perfectly clear and honest, what I know works best adjusting the PID variables is pushing the "AUTO TUNE" button – at least that usually doesn't make it worse.

According to Automation Directs Chapter 8 of the DL05 Micro PLC User Manual, the control output at time "t" is:

t

M(t) = Kc[ e(t) + 1/Ti∫0 e(x) dx + Td d/dt e(t) ] + M₀

Where :

Kc = proportional gain

Ti = Reset or integral time

Td = derivative time or rate

SP = Setpoint

PV(t) Process Variable at time "t"

e(t) = SP-PV(t) = PV deviation from setpoint at time "t" or PV error.

Now, to relate this to an action the operator can do, to:

Eliminate the Integral Action (so-called Reset), set Ti = 9999

Eliminate the Derivative Action (so-called Rate), set Td = 0

Eliminate the Proportional Action (so-called Gain), set Kc = 0

(according the them)

Mathematically, PID control loops get very specific to the system, the actuators and the sensors. In fact, I think the actuators are way more important than the math.

There probably is no 'general' case, but this is a differential equation. Right?

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#26
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Re: Ordinary Differential Equations – A Review of Basic Solutions

09/05/2010 8:06 PM

I have always found that there are two types of engineers,

1.) theoretical

2.) practical

having run into theoretical engineers who would spend days on a problem on a practical application, that would not make one bit of difference wither way....and end up getting it wrong anyways. I asked them if they would like my advice. Their response is, yes.......I would consider it.

My advice is...and this is a quote...."Not to analyze the shit out of something until its unrecognizable, accept it as it is and move on."

As a added note, this is a very good post that I have been following (i.e. refreshing) with great interest.

p911

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