Question

Can I get some help solving this Discrete-Time Dynamical System?

The updating function is f(x)=.75x=2

and I've worked it out to be

f(5)=.75^5(16)+2(.75^4)+2(.75^3)+2(.75^2)+2(.75)+2

But I can't figure out how to solve that to be the full solution. Any thoughts?

Answer 1

\(m_t=0.75m_{t-1}+2\)
Above is our difference equation, but to solve, we need more information... what is the initial value(s)?

Answer 2

\[f(5)=.75^5(16)+2(.75^4)+2(.75^3)+2(.75^2)+2(.75)+2\]

is what it really looks like. I just figured out the equation editor.

Answer 3

M(0)=16

Answer 4

So, using M(0)=16 and the updating function f(x)=0.75x+2 , I'd like to find the solution.

Answer 5

Uh.. what is that \(f(5)\) for? That is neither the updating function nor is it an initial value...

Answer 6

So, I should have typed \[M_5. \] = that whole mess.

Answer 7

Well, let's solve the more general \(m_{t+1}=am_t+b\)... given \(m_0=c\), we have:$$m_1=ca+b\\m_2=a(ca+b)+b=ca^2+ab+b\\m_3=a(a^2c+ab+b)+b=a^3c+a^2b+ab+b\\...\\m_t=ca^t+b\sum_{k=0}^{n-1}a^k$$

Answer 8

Now that we've figured that out, we can see our updating function is a special case where \(a=0.75\), \(b=2\), and \(c=16\). We can write our solution:$$(m(t)=16a^t+2\sum_{k=0}^{t-1}0.75^k$$

Answer 9

Ok. That makes sense. I guess I couldn't figure out the way to express that solution. Because I saw the pattern going on, but couldn't figure out how to express it. Could you explain what the numbers around the summation symbol mean?

Answer 10

Which numbers? The one out front is merely a coefficient... see how in \(m_3\) you have \(a^2b+ab+b=a^2b+a^1b+a^0b=b(a^2+a^1+a^0)\)?

Answer 11

Yeah, so I see that pattern. I'm confused about how the sigma annotation, and how that would be computed. So, like, literally, how would I take
\[M(t)=16(.75^t)+2\sum_{k=0}^{t-1}0.75^k\]
and put it in a calculator?

Answer 12

Do you have a graphing calculator? Most support summation.

Answer 13

Yeah, I have a TI83.

Answer 14

Sorry, I didn't really know you wanted a totally closed-form polynomial function... it's weird how similar difference and differential equations are, though! The solution should be:
$$M(t)=8(0.75)^t+8$$

Answer 15

http://mtl.math.uiuc.edu/node/87

Answer 16

Hmmmm... Well, that equation looks a little better than the sigma one. Can you explain how you got it?

Answer 17

http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/discrete_ch14.pdf
Sure! We're dealing with a first-order linear non-homogeneous difference equation with constant coefficients. The idea is that we can simplify a geometric progression rather easily.
$$M(t)=16(0.75)^t+2\sum_{k=0}^{t-1}0.75^k\\\ \ \ \ \ \ \ =16(0.75)^t+2\left(\frac{1-0.75^t}{1-0.75}\right)\\\ \ \ \ \ \ \ =16(0.75)^t+8\left(1-0.75^t\right)\\\ \ \ \ \ \ \ =16(0.75)^t+8-8(0.75)^t\\\ \ \ \ \ \ \ =8(0.75)^t+8$$

Answer 18

http://en.wikipedia.org/wiki/Geometric_progression#Geometric_series
$$\sum_{k=0}^nr^k=\frac{1-r^{n+1}}{1-r}$$

Answer 19

Sheesh, that is perfect. Thank you so much for all your help!

I hadn't even heard of geometric progressions until just now. I don't know why my teacher didn't let us in on that one before assigning this.

Thank you so much.

Answer 20

No problem! Out of curiosity, what year of your university studies are you taking this in?

Answer 21

Calculus 246 (It's Calculus 1 for Biology Majors.)

My teacher might be assigning hard stuff, or I might just be behind. I took the prerequisites over a year ago... So that might be it.

But thanks again.

Answer 22

Neat! Are you a freshman? I'm a junior in high school so I'm trying to gauge how long I have to learn this stuff :-p

Answer 23

Sheesh, nope. I'm a junior. I think a lot of the people in my class are freshmen straight out of high school though.

I took Precal in my Junior Year in high school, Calculus in my Senior year, and then I took a few years off to travel before college. So when I jumped back in, I took Math 111 and Math 112 (College Algebra and Trig) and now I'm in Math 246. So, it's been a year since I took 112, and about 4-5 since I was deep into calculus.

Are you planning on going straight through to college? If so, they'll probably let you take a placement test, and if you're math is fresh in your mind, you can probably save yourself a lot of tuition/time by getting into a more advanced math class. Good for you!

Answer 24

(I am totally bummed that I didn't keep my math skills fresh and carry them straight through to college. If you can do that, you'll save yourself some frustration! :) )

Answer 25

BTW, being taught by a high school junior simultaneously makes me feel dumb, and inspired by the internet/this website/high school kids. Keep up the good work! :)

Answer 26

Yeah, I don't think I'll be taking any gaps... I'm taking as many math classes as I can before I graduate next year (Diff. EQ, lin. alg., etc.)! Don't consider yourself as being taught by me, I just tried to help you out a little... besides, I'm a math tutor at a local community college where I take courses!

Answer 27

Smart plan. You're awesome. Good for you! :)