2nd order differential equation...anyone?

[Robert]

I've noticed some brainteasers floating around CA lately.  Personally, I say we "man up" (or "woman up") and do something that requires a bit more brainpower.  With this in mind I have a calculus problem.   Anyone?

y'' - y' - 6y = 0

Where y'' is the second derivative; y' is the first derivative and y is the original function. 

You need to solve for y given the following:

y = f(t)

y(0) = 1

y'(0) = 2

[Robert]

If you need a clue to get started I'll give you the following:

y = c1*e^(r1*t) + c2*e^(r2*t) 

This is read as follows:  

The original function (y) equals the first constant (c1) times the natural base (e or 2.718...) to the power (^) of the first root (r1) times time (t) plus the second constant (c2) times the natural base (e or 2.718...) to the power (^) of the second root (r2) times time (t).

Also, keep in mind the derivative of e^u (where "u" is a differentiable function) is expressed as follows:

d(e^u)/dt = e^u*du/dt

[Outta Here]

Sorry. Even though I have an Bachelor's in Accounting and an Associate's in Math, it's been too long since I've had any need for this. Can't do it. Here's to anyone who's been able to maintain their knowledge through the years--good on you!

P.S., if you're looking for a real challenge to "man" up to...spend a lifetime caring for other people so much so that there's no time in the day for individual pursuits.

 

[Robert]

[Robert]

Perhaps  David Geelan will be dropping by?  Maybe he uses differential calculus in his line of work?

 

 

[rudywoofs (Pam)]

I avoided calculus like the plague.  But I loved statistics.

[Gregory Matthews]

Calculus is beyond my knowledge.

 

[Outta Here]

I asked my daughter who has a Master's in Actuarial Science...she says, "I dunno. That was 6 years ago and we don't ever use it."

[Robert]

11 hours ago, Robert said:

I've noticed some brainteasers floating around CA lately.  Personally, I say we "man up" (or "woman up") and do something that requires a bit more brainpower.  With this in mind I have a calculus problem.   Anyone?

y'' - y' - 6y = 0

Where y'' is the second derivative; y' is the first derivative and y is the original function. 

You need to solve for y given the following:

y = f(t)

y(0) = 1

y'(0) = 2

1]  Let y = e^(rt)

2]  compute the derivative of y with respect to time (t) using the formula: d(e^u)/dt = e^u*du/dt

let u = rt, therefore d(e^(rt))/dt = e^(rt)*d(rt)/dt or y' = r*e^(rt)

3] Compute the derivative of dy/dt with respect to time (t) or y''

let u = rt, therefore d(dy/dt)/dt = y'' = d(r*e^(rt))/dt or r*e^(rt)*d(rt)/dt or y'' = r^2*e^(rt)

4] Now substitute y, y' and y'' back into the original problem:

r^2*e^(rt) - r*e^(rt) -6*e^(rt) = 0

5] Factor out e^(rt)

e^(rt)*(r^2 - r -6) = 0

therefore r^2-r-6 = 0

6] Factor the quadratic

(r + 2)(r - 3) = 0

7]  Recall vector space formula:   y = c1*e^(r1*t) + c2*e^(r2*t) 

8] Now compute the roots,  r1 & r2

r1 = 3 

r2 = - 2

9] Enter this data back into the vector space formula of step 7

hence, y = c1*eC^(3t) + c2*e^(-2t)

10] Now we must calculate c1 and c2

To do this we need to use the original parameters, which are:

y(0) = 1

y'(0) = 2

11] First let's use y(0) = 1 in the formula derived in step 9

y(0) of c1*eC^(3t) + c2*e^(-2t) = c1+ c2 = 1

So y(0) = 1 =  c1+ c2

time for a break....

 

 

[Robert]

3 hours ago, Gregory Matthews said:

Calculus is beyond my knowledge.

 

spelling without spell check is beyond my knowledge....

[Robert]

7 hours ago, Robert said:

1]  Let y = e^(rt)

2]  compute the derivative of y with respect to time (t) using the formula: d(e^u)/dt = e^u*du/dt

let u = rt, therefore d(e^(rt))/dt = e^(rt)*d(rt)/dt or y' = r*e^(rt)

3] Compute the derivative of dy/dt with respect to time (t) or y''

let u = rt, therefore d(dy/dt)/dt = y'' = d(r*e^(rt))/dt or r*e^(rt)*d(rt)/dt or y'' = r^2*e^(rt)

4] Now substitute y, y' and y'' back into the original problem:

r^2*e^(rt) - r*e^(rt) -6*e^(rt) = 0

5] Factor out e^(rt)

e^(rt)*(r^2 - r -6) = 0

therefore r^2-r-6 = 0

6] Factor the quadratic

(r + 2)(r - 3) = 0

7]  Recall vector space formula:   y = c1*e^(r1*t) + c2*e^(r2*t) 

8] Now compute the roots,  r1 & r2

r1 = 3 

r2 = - 2

9] Enter this data back into the vector space formula of step 7

hence, y = c1*eC^(3t) + c2*e^(-2t)

10] Now we must calculate c1 and c2

To do this we need to use the original parameters, which are:

y(0) = 1

y'(0) = 2

11] First let's use y(0) = 1 in the formula derived in step 9

y(0) of c1*eC^(3t) + c2*e^(-2t) = c1+ c2 = 1

So y(0) = 1 =  c1+ c2

 

12] Now we need to figure y'(0) = 2 using the vector space formula of step 9

So we need to find the derivative of the vector space formula or d(c1*eC^(3t) + c2*e^(-2t))/dt 

dy/dt = y' = d(c1*eC^(3t))/dt + d( c2*e^(-2t))/dt = 3*c1*e^(3t) - 2*c2*e^(-2t)

13] Now take y'(0) = 2

2 = 3*c1 - 2*c2

14] Now we have two unknown variables (i.e., a simultaneous equation)

1 =  c1+ c2

2 = 3*c1 - 2*c2

Solving for c1 

c1 = 4/5

solving for c2

c2 = 1/5

14] substituting back into the vector space formula we have the original function:

y = (4/5)*e^(3t) + (1/5)*e^(-2t)

Now, only if I could spell half decent....

[Robert]

Okay, maybe something easier than calculus.  How about functional trigonometry?

Solve for y

y = Sin(2arcSin(1/7)) 

 

[Robert]

Hint:  Use the identity, Sin2x = 2SinxCosx

[Outta Here]

Robert, I don't think anyone here except for maybe Bravus can do this...not anymore, anyway. Lots of us may have learned in the past, but we don't use it on a regular basis, so it's "not fresh".

[Robert]

Then is our educational system really teaching us or are they teaching us to take test without really understanding the material?  In my estimation it's the latter.  Short term memorization and then regurgitate that info for a test.  This is what's wrong with American educational system. 

Apparently the Chinese teach differently because my co-worker, who did not major in mathematics, can do basic calculus to this day.  

My two cents.

[Outta Here]

Are you saying that everything you've ever mastered, you've retained?? I've mastered quite a lot in my education...really and truly understanding the material...but don't ask me to reproduce it 30 years later....a lot of other more important information took up my brain activity. The Calculus is still in there....somewhere....but the neural pathways are clouded and directionally confused.

How old was your coworker when he learned it? Does he have children? Is there some reason he would have retained it more readily than the average person you're working with?

[Outta Here]

Could it possibly be because he is Chinese and learned Calculus in a completely different way...think Malcolm Gladwell? 

Did he also learn other subjects for mastery...biology, chemistry, physics, A&P, English....DOS, Pascal, Cobol...has he retained all this as well?

[Gregory Matthews]

Robert:

The question as to what we  should teach our children has been debated for many years.

Dr. A. T. Benjamin, of Harvey Mudd college has specialized in teaching people how to do math problems, in their heads, and almost instantly.  As part of his teaching, he teaches his students how to do Vedic Mathematics.   E.G.  He might ask a student to divide 932,498 by 83.  The properly trained student would quickly come to the correct answer of 11,234 with a remainder of 76.

But, Robert, does the fact that we have probably not taught our students Vedic Division  and thererfore they are unable to calculate the above answer in their heads make them any less able to function in the world in which we live?  I would not want to judge the effectiveness of our math courses by whether or not could quickly calculate the above problem in their head. 

[JoeMo]

Not everyone needs to know calculus or trig to perform efficiently in their jobs.

[Robert]

"As a college professor, I see the dysfunctional effects of an educational system based on testing when I look out into a room full of students. After years cramming, memorizing, regurgitating, and forgetting, many students enter college with little intellectual curiosity much less a sense of academic excitement. Too often, the students just want to be told what they need to learn to pass the test or what they need to write to get a good grade on a paper. Because so much of their schooling has been based on this dysfunctional model, they have forgotten how to be the self-directed and genuine learners that they were when they first entered school. "

 

Here's the complete blog:  http://www.everydaysociologyblog.com/2012/04/cram-memorize-regurgitate-forget.html

[Bravus]

LOL - clearly haven't been swinging by frequently enough to sniff the bait.

I certainly *used* to know how to solve second order ODEs (ordinary differential equations) when I did my Graduate Certificate in Science back in 2011-2012, but it does tend to be a bit 'use it or lose it' when you're not doing constant practice. I could *follow* Robert's solution, but I'm not sure I could have come up with it on my own now.

Fun stuff though. I'm more scared of the trig... which is a weakness I really should learn enough to conquer.

Suzie talked me down off the ledge of trying to dive back into a Masters in physics in the midst of the rest of my busy life this week. I think that really will have to wait until retirement.

[Bravus]

I've been playing on Socratic.org lately, which is a site that allows students to ask questions and get answers. I always try to answer so that they learn the concept, not just write down the answer to their homework. 

It has a really nice simple markup system for math/science notation, which makes it look neater and more authentic than is manageable in an environment like this.

[Bravus]

 

[Outta Here]

Yep. That looks right!

[Gregory Matthews]

Who is Dr. A T Benjamin?

 

Quote

Arthur T. Benjamin

Smallwood Family Professor of Mathematics

 

Ph.D., The Johns Hopkins University

 

Research Interests: Combinatorics, game theory, operations research

 

Shanahan 2414 • 18688/(909) 621-8688 • benjamin@math.hmc.edu