Anyone good at differential equations.

[Looneybomber]

We're doing 2nd order nonhomogeneous equations. Because of break, I don't have class again till Thursday.

The problem:
y''-2y'-3y=3te^(2t), y(0)=1, y'(0)=0

Because it's nonhomogeneous, we need to make it homogeneous in order to solve it, that is y''-2y'-3y=0

To make it homogeneous, we need to find a value of Y that we can add to the general equation in order to make it homogeneous. In most cases, you usually replace the coefficient of the resultant with a substituted coefficient, A. For example, y''-3y'-4y=3e^(2t), so we'd use Y(t)=Ae^(2t), then solve for Y'(t) and Y''(t), so that Y''(t)-3Y'(t)-4Y(t)=3e^(2t)

So, back to my problem, naturally I picked Y(t)=Ate^(2t), replacing the coefficent with A, however, the correct function to use was Y(t)=Ate^(2t)+Be^(2t). Why did they pick the function that they did? I know if you have a resonant response, you add an extra t, for example, y(t)=e^-t would result in a function of Y(t)=Ate^-t, but is this resonant?

The general solution is y(t)= C1e^3t + C2e^-t - te^2t - 2/3e^2t (with spaces added between terms) where C1 and C2 are arbitrary constants. It was an initial value problem and we later solved for C1 and C2.

[themadscientist]

4.11:1 is great for drag racing, but sucks on the highway.

[alms24sebring]

I hate calculus and not being able to understand it, even though Ive taken 3 classes.

[Looneybomber]

I don't mind calc since it's usable in physics and circuits, and I prefer calc over algebra (unless it's linear algebra). I've had 2 classes (didn't have to take vector calc for my major) for 10hrs total. I almost decided to minor in math, but I didn't want to add two more 500 level classes this semester. Bleh. Diff eq is causing me enough problems.

[alms24sebring]

God I cant even imagine whats in a 500 Math. This is this part of this forum that always makes me wanna throw up everywhere whenever I go into it just browsing. This is a very in depth forum though with a bunch of masters at math and physics. Since I still dont know what the OP is asking, I hope this will help in your search on finding the square root of the meaning of life.

http://www.physicsforums.com/forumdisplay.php?f=4

[s0m3th1ngAZ]

square root of the meaning of life.

42

[C-Kwik]

We're doing 2nd order nonhomogeneous equations. Because of break, I don't have class again till Thursday.

The problem:
y''-2y'-3y=3te^t, y(0)=1, y'(0)=0

Did you leave out a 2 in the exponent? The particular part of the solution you posted doesn't make sense otherwise.

So, back to my problem, naturally I picked Y(t)=Ate^(2t), replacing the coefficent with A, however, the correct function to use was Y(t)=Ate^(2t)+Be^(2t). Why did they pick the function that they did? I know if you have a resonant response, you add an extra t, for example, y(t)=e^-t would result in a function of Y(t)=Ate^-t, but is this resonant?

What you actually have is a first degree polynomial times exp(2t). So you need a coefficient for each of the terms. Consider if the function on the right side was (t^3 - 2t^2 + 4t - 2) exp(2t). Then the particular solution would take the form Yp = (At^3 + Bt^2 + Ct + D) exp(2t). In your case, think of it as looking at this: (3t + 0) exp(2t). Inserting the t in the case of resonance is necessary because the coefficient A is not solvable (A = 0). I can see the confusion though , but its just an entirely different rule.

I'll also refer you to this:

http://www.math.psu.edu/tseng/class/Mat ... %20pt2.pdf

It goes over non-homogeneous linear differentials in general and is pretty helpful. And on page 21, there is a small chart that might help you with how to determine what to use for Yp.

[float_6969]

Dustin, you haz lots of smarts.

[Looneybomber]

We're doing 2nd order nonhomogeneous equations. Because of break, I don't have class again till Thursday.

The problem:
y''-2y'-3y=3te^t, y(0)=1, y'(0)=0

Did you leave out a 2 in the exponent? The particular part of the solution you posted doesn't make sense otherwise.

Good eye, I did leave out a 2! I'll go fix that.

What you actually have is a first degree polynomial times exp(2t). So you need a coefficient for each of the terms. Consider if the function on the right side was (t^3 - 2t^2 + 4t - 2) exp(2t). Then the particular solution would take the form Yp = (At^3 + Bt^2 + Ct + D) exp(2t). In your case, think of it as looking at this: (3t + 0) exp(2t). Inserting the t in the case of resonance is necessary because the coefficient A is not solvable (A = 0). I can see the confusion though , but its just an entirely different rule.

And there's my problem, I ignored that I had a polynomial and focused solely on the exp.
I think you've helped me with digital logic (boolean algebra) and now diff eq. Thanks.

[Looneybomber]

Dustin, you haz lots of smarts.

I try to look the part, without actually playing the part. That's why I get on here and wolfram alpha for help. But thanks man!

What I do have is a lot of beer! Just bottled 10 gallons on Sunday (gotta be around 30gal bottled now). Sadly though, I haven't had time to BREW another batch, IE, your chocolate stout.

[C-Kwik]

And there's my problem, I ignored that I had a polynomial and focused solely on the exp.
I think you've helped me with digital logic (boolean algebra) and now diff eq. Thanks.

No problem. Though, its self serving actually. Helping others with this helps reinforce my math skills. I hadn't touched this stuff in a while (we usually use Laplace transforms) so its great review.

[Looneybomber]

And there's my problem, I ignored that I had a polynomial and focused solely on the exp.
I think you've helped me with digital logic (boolean algebra) and now diff eq. Thanks.

No problem. Though, its self serving actually. Helping others with this helps reinforce my math skills. I hadn't touched this stuff in a while (we usually use Laplace transforms) so its great review.

That's what everyone else in my major says; Laplace is when diff eq starts getting useful. Our diff eq teacher mentions the same thing. 4th order ODE's is when he says diff eq gets useful, but that's because he's a physics PhD and talks about beam deflection in bridges and other things. But you gotta start somewhere right?

[alms24sebring]

Well look what I just stumbled across with good timing.

[float_6969]

Dustin, you haz lots of smarts.

I try to look the part, without actually playing the part. That's why I get on here and wolfram alpha for help. But thanks man!

What I do have is a lot of beer! Just bottled 10 gallons on Sunday (gotta be around 30gal bottled now). Sadly though, I haven't had time to BREW another batch, IE, your chocolate stout.

Yea, I'm really wanting to get over there and hang out, but I'm super busy with work right now. Work should slow down by the end of the month and I should have a lot more free time.

[PapaSmurf2k3]

I recall hating Diff EQs when I took it, and I really have no desire to re-visit that. Beam deflection was easy as hell, provided you have the formulas you need. Laplace and fourier transfers are rarely used as well.

Calc is pretty useless unless you HAVE to derive your own formula for some reason. There are very few physics formulas that are used in real world situations that require calc (one of the only ones I can think of is the car-accident scenario, where a car is stopped in the road, oncoming dude starts braking and stopped car starts accelerating... where do they crash [if they crash]).

[C-Kwik]

I recall hating Diff EQs when I took it, and I really have no desire to re-visit that. Beam deflection was easy as hell, provided you have the formulas you need. Laplace and fourier transfers are rarely used as well.

Calc is pretty useless unless you HAVE to derive your own formula for some reason. There are very few physics formulas that are used in real world situations that require calc (one of the only ones I can think of is the car-accident scenario, where a car is stopped in the road, oncoming dude starts braking and stopped car starts accelerating... where do they crash [if they crash]).

Curious. Is that true for controls engineers too? I'm starting my second controls class and it sounds like diff equations using Laplace is useful for understanding and predicting system behavior.

[PapaSmurf2k3]

Yes. You will very rarely encounter a system that you have to use Diff EQs to predict. Most times you can just rely on your engineering background to say "we need more of this", or "this" based on some sort of experimental outcome. I'd venture to say every company worth a s*** makes you validate whatever math you'd be doing by building a physical prototype or 50.

[MinisterofDOOM]

4.11:1 is great for drag racing, but sucks on the highway.

Glad I'm not the only one who thought this.

My favorite calculator for differential equations:

http://www.grimmjeeper.com/gears.html

[C-Kwik]

Yes. You will very rarely encounter a system that you have to use Diff EQs to predict. Most times you can just rely on your engineering background to say "we need more of this", or "this" based on some sort of experimental outcome. I'd venture to say every company worth a s*** makes you validate whatever math you'd be doing by building a physical prototype or 50.

I did a brute force method of putting together a PID control for a balanced inverted pendulum a couple of years ago. It wasn't hard to do but was tedious without a good starting point. I have to wonder if I could do it faster by modelling it. Does your experience in this also include high precision controls (medical equipment, high precision mills, etc.)? There are lots of medical equipment companies in my area so it may be important for me to consider what they are looking for in this aspect.

[Looneybomber]

It's off topic, but I've looked into PID's, for controlling a heating element for a larger automated home brew setup.
Are you a mech-E student/graduate?

[PapaSmurf2k3]

I did a brute force method of putting together a PID control for a balanced inverted pendulum a couple of years ago. It wasn't hard to do but was tedious without a good starting point. I have to wonder if I could do it faster by modelling it. Does your experience in this also include high precision controls (medical equipment, high precision mills, etc.)? There are lots of medical equipment companies in my area so it may be important for me to consider what they are looking for in this aspect.

Yeah, you have to be in an extremely specialized field/application to even graze the surface with that stuff. I suppose if you're the guy making the CNC mill, you might use it? Then again I can't even really think why, you'd just use a stepper motor or encoder or something to tell distance/position. Everything after that is just G-code, which doesn't use Diff EQs at all. Not to mention most programs are already based on some sort of software that already exists by this point. You really have to be on the "inventing" side of very specialized things to use it.

I have a degree in Mechanical Engineering and German.
http://www.uri.edu/iep/ <--- my program. Graduated in 2008. 100% job placement after graduation is no joke.

[C-Kwik]

It's off topic, but I've looked into PID's, for controlling a heating element for a larger automated home brew setup.
Are you a mech-E student/graduate?

Mech-E student.

Yeah, you have to be in an extremely specialized field/application to even graze the surface with that stuff. I suppose if you're the guy making the CNC mill, you might use it? Then again I can't even really think why, you'd just use a stepper motor or encoder or something to tell distance/position. Everything after that is just G-code, which doesn't use Diff EQs at all. Not to mention most programs are already based on some sort of software that already exists by this point. You really have to be on the "inventing" side of very specialized things to use it.

Gotcha. My most practical exposure to it was through the grad student I was working for in an internship this summer. Though, we ended up not getting really deep into it anyways as other priorities took over and they pulled me into the priority project. I was kinda hoping to get more exposure to it. My thoughts would be they would likely seek Masters degree holders for that kind of work, but it could never hurt to know a little more about it.

I was using the CNC mill as an example for indicating precision, but high precision medical equipment, (say some kind of surgery robot or laser) might require fast response with minimal overshoot. That said, the biggest concern in general just seems to be making sure the control doesn't start getting out of control. I suspect that would not require much heavy math and with experience, would become somewhat intuitive in most scenarios. Either way, I'm just trying to get a feel for the practical use of this stuff...Thanks!

[PapaSmurf2k3]

Lots of the high precision stuff I've used (robots and press machines) just use VERY simple algorithyms to get where they go. Something like "I'm here, I want to go here (within some tolerance). Go 98% there, check your position, repeat as necessary until I'm in tolerance".