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anonymous
 one year ago
how exactly does the standard form of the exponential function
y[x] = k E^(r x)
relate to this form of the logistic differential equation
y'[t] == r y[t] (1  y[t]/b)
and it's counterpart
y[t] = (b*E^(r*t + b*k))/(1 + E^(r*t + b*k))
r is obvious
k is not so obvious but probably the same between y[t] and y[x]
but b???!!! b is a problem.
anonymous
 one year ago
how exactly does the standard form of the exponential function y[x] = k E^(r x) relate to this form of the logistic differential equation y'[t] == r y[t] (1  y[t]/b) and it's counterpart y[t] = (b*E^(r*t + b*k))/(1 + E^(r*t + b*k)) r is obvious k is not so obvious but probably the same between y[t] and y[x] but b???!!! b is a problem.

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UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0omg I feel like I'm in Mathematical Biology again

jagr2713
 one year ago
Best ResponseYou've already chosen the best response.0@nincompoop @Australopithecus

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0lol, its awful, Im in hell

anonymous
 one year ago
Best ResponseYou've already chosen the best response.03 days I been trying to work this out

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0where there are two parameters in the logistic differential equation... r is the growth rate k is carrying capacity. ewww I don't want to do this again!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0you any good with mathematica usuki?

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0but I know this logistic equation

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0In all the examples they provide b.. but then in the problem.. they dont.. I do have a table of data though.. an exponential sample.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0is that required to work out b in the final equation?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0my best guess is that the logistical curve is going to be an S .. at the middle of that S, in the 2nd derivative will be a 0 marking the point at which the concave changes.. this point is going to be at almost exactly 1/2 of b.. if there is proven to be a relationship between the corresponding exponential curve and the logistical curve then I can find b.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0like the slope of the derivatives, might have some relationship that is propotional to r.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0at that inflection point.

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0What exactly is the problem, this seems like logistic differential equation, I mean like does k represent the carrying capacity, and r,k >0 constant?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0My problem is that I was given the top equation.. and asked to turn it into the bottom equation. after they showed me the middle equation.. and they're asking me to work it out.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0if we name the equations 1,2,3. I know how to go from equation 2 to 3 .. but not from equation 1 to 2 or 1 to 3.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0k as far as I can tell, is the value where x=0 y[0], I do have a value for it.. I have a data table for the exponential points.. and a formula that approximates those data points.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the formula is in the standard exponential form [x] = k E^(r x)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oops y[x] = k E^(r x)

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0The counterpart to me seems to be integrated, but I'm not really used to this stuff yet (haven't done much ODE atm), I've done similar but for word problems haha..mhm. I would take an attempt at it, but I don't want to lead you to the wrong place so I'll let the big guns handle it, so to speak haha @ganeshie8 @SithsAndGiggles

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0thank you astro for taking a look. appreciated.

jagr2713
 one year ago
Best ResponseYou've already chosen the best response.0Sorry about the inconvenience @hughfuve i dont want to make false promises but i will try and find someone to help :D @freckles

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0I've done logistic equation... but like I mentioned I don't want to go back to Mathematical Biology >_<

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well I appreciate all help.. no matter how it turns out..

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0y'[t] == r y[t] (1  y[t]/b) there are two parameters r and b. r is the growth rate and b is the carrying capacity.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0you can't assign values to parameters.. the derivative is a constant.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so if b is the carrying capacity here.. could it just k from the first equation? The problem I have is when I tried that I got a very small S curve.. the data I have represents the US population, and according to a value of b=k, then the US population should not go over 80million.. as that is clearly not the reality.. I am concerned.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0hmmm mine had ecology situations

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0like for bird populations

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0lol, you studied at the height of environmental hysteria

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0this course one comes from a time of immigration hysteria

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0my Mathematical Biology book is still in my backpack since final exam week. I'm not taking it out XD!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0touching it probably makes your fingers rash

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0my book is going to give me shingles one day.. I just know it.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0well it's not a discrete model (that's for difference equations) it's a continuous model.. what I 'm saying may be a different point of view for the logistic equation

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes this is one of the first problems of my chapter introducing differential equations. what an intro.. geeez.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0pfffffffft you should be doing integrating factor as the first topic

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0forgot to mention that the logistic equation is separable _!

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0Does that look right Hugh?

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0dw:1435638707408:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I am on chapter 6.. integration isn't until chapter 10. I have a hunch that perhaps this course is not for teaching calculus, but instead for teaching mathematica to those who already know calculus.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0your curriculum is backwards _

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0Populations tend to get larger until there is no longer enough food or space to support so many individuals. This type of growth is called logistic population growth What Is Logistic Population Growth? A group of individuals of the same species living in the same area is called a population. The measurement of how the size of a population changes over time is called the population growth rate, and it depends upon the population size, birth rate and death rate. As long as there are enough resources available, there will be an increase in the number of individuals in a population over time, or a positive growth rate. However, most populations cannot continue to grow forever because they will eventually run out of water, food, sunlight, space or other resources. As these resources begin to run out, population growth will start to slow down. When the growth rate of a population decreases as the number of individuals increases, this is called logistic population growth.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I have the textbook calculus concepts and contexts by james stewart.. I just found a section in there on this formula.. Its almost word for word

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0James Stewart? I had a Single Variable Calculus Book from that author, but it was only Calculus I and II.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0logistic equation is a 100% ODE though

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0Is this in james ste 7e?!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I got the larson hostetler, edwards video seriies too.. maybe theres something in there.

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0james stewart 7th edition

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh no.. its 2nd edition

anonymous
 one year ago
Best ResponseYou've already chosen the best response.02001, it was on our bookshelf.. my g/f used it way back when

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0lol, no wonder Im struggling.. I need some new stuff..

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I have pearsons college mathematics 13th edition if that helps

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0but they dont say much on this topic

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0actually this looks promising http://reference.wolfram.com/language/tutorial/DSolveOverview.html

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0might have found a possible solution, just have to work it backwards. if y[x] = logarithmic S curve f[x] = corresponding exponential curve with same r then y''[n]==0 solve for n finds the mid point of the S curve. Then take that n of midpoint of S curve and f[n]+f[0] = value for b

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0well well y[n]/2 =f[n]+f[0]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0might have found a possible solution, just have to work it backwards. if y[x] = logarithmic S curve f[x] = corresponding exponential curve with same r then y''[n]==0 solve for n finds the mid point of the S curve. Then take that n of midpoint of S curve and f[n]+f[0] = value for b well well y[n]/2 =f[n]+f[0] so (y[n]/2)f[0] =f[n]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0You are right about the logistic equation producing scenarios that can't happen. Like the one for your 80 million people in the US. Obviously to sustain 80 million, you have to force people to not have plenty of kids. It's like the logistic equation is saying ok your world must sustain exactly 80 million people or less and if it goes 1 over, you'll have a shortage of supplies. Same with the birds. Either 100 birds or less...otherwise if there are 1000 birds there will be less resources and eventually the birds will die.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so does that mean then that b is arbitrary? and you can set it to whatever you want? say if you determine that you are going to put some cap on on growth and force it?

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0you can try to force it .. but in reality that's not possible... humans are going to mate anyways... same with birds.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0@Michele_Laino is here!

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0I was told that Mathematical Biology is harder ... it actually takes Calculus III, IV, ODE, and Linear Algebra... puts it in a blender and messes everything up.

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0plus I had to use Matlab for assignments and a project... I still think my project about Killer Zombies saved my grade.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0maybe I got thrown off because the examples show exponential plots vs logarithmic plots that are proportional to each other in respect to the value for b. They created a log plot, then used that to make the exponential plot. But now I am thinking that maybe the two plots are always proportional in this way no matter what the value for b.. and b is arbitrary, and it depends on what controls you want to place on the model. So Im looking for a shadow

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0crap.. I think it is the case

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3if we try this substitution: \[y\left( t \right) = \frac{1}{{z\left( t \right)}}\] where z is the new variable, then our starting equation, namely equation #2 can be rewritten as follows: \[\frac{{dz}}{{dt}} + rz = \frac{r}{b}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Hi michele.. welcome :)

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0now it looks like the integrating factor in the form of \[\frac{dy}{dx} =p(x)y=q(x)\]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0NUGH NOT MY NIGHT! \[\frac{dy}{dx} +p(x)y=q(x)\]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0I can't do math in the heat D:!

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3the solutions to that ODE are: \[\Large z\left( t \right) = {e^{  rt}}\left( {c + \int {\frac{r}{b}{e^{rt}}dt} } \right)\]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0so you do need integrating factor right?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3right! so we have: \[\Large z\left( t \right) = c{e^{  rt}} + \frac{1}{b}\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3now we have to return to our old variable y(t): \[\Large y\left( t \right) = \frac{b}{{bc{e^{  rt}} + 1}}\] where c is an arbitrary real constant

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0@hughfuve didn't you mention earlier that integrating factor was not until real later in your book?

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0"hughfuve Best Response Medals 1 I am on chapter 6.. integration isn't until chapter 10. I have a hunch that perhaps this course is not for teaching calculus, but instead for teaching mathematica to those who already know calculus."

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes, but that's okay.. I can still use the Iintegrate[] mathematica command in this problem at least..

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0I would complain to your instructor if I were you. How can you do this problem without integrating factor?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Its okay.. Im learning for now.. this will at least put me in a place where I can put this one to rest..

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0.___. last time I checked all of differential equations come first...then the Mathematical Biology comes second

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3now we can multiply both numerator and denominator by \[\Large {{e^{rt}}}\] so we get: \[\Large y\left( t \right) = \frac{{b{e^{rt}}}}{{bc + {e^{rt}}}}\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3now we have to know the initial conditions, since the formula above gives us infinite solutions

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so the exponential function I started with.. r = 0.01300265909581666; k = E^(4.382968909053888); yExp[x] = k E^(r x) I can basically pull the variables here as the starting conditions and then b is whatever we want to set the limit of the function too?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I assume c =k r = r b = whatever

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0and I will have an exponential version and a logarithmic version that start out the same, but end up going in different directions.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0hence the idea that.. logistical growth is controlled growth ?

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3the coefficient b is called "carrying capacity"

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ah gotcha, thanks.. the language helps

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3it is suffice to use this substitution: \[\Large \frac{1}{c} = {e^{bk}}\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3I have integrated that ODE using the separation of variable method, and I got this solutions: \[\Large y\left( t \right) = \frac{{b{c_1}{e^{rt}}}}{{1 + {c_1}{e^{rt}}}}\] where c_1 is the integration constant. Now it is suffice apply this substitution: \[\Large {c_1} = {e^{bk}}\]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.0yes that's correct... though these days I prefer integrating factor... I know it's more steps but taking the antiderivative is so much easier than integrating on both sides with separation of variables and end up getting stuck on the left side.
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