The Logistic Equation and the Analytic Solution
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- เผยแพร่เมื่อ 30 ก.ย. 2024
- Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) / patrickjmt !! The Logistic Equation and the Analytic Solution. In this video, I find the analytic solution to the logistic differential equation.
![Exact Equations [ODE]](http://i.ytimg.com/vi/pgJci5CI9n4/mqdefault.jpg)
Dude, At Khan Academy Sal tries to show how to solve the thing, too. However, for some reason he is not being all that clear about it and, according from what can be picked up from the comments, many people are left confused. You are doing so much better of a job here, so I suggest you send in this clip as a basis for improvement to K.A.!
Perhaps, but the Khan end product is the generally used logistic formula. Patrick's logistic formula
is likely consistent with the general representation, but I haven't been able to demonstrate it yet
I also decided to search for an alternative way of finding the analytic solution for the logistic differential equation because Sal's explanation in Khan Academy was really confusing. I'm glad I found this video, it's explained really well here.
I watched that one too, and thought the work was good, but it is another rats and rabbits curve, not %votes in elections 1, 2, 3, 4... n
That does it. Instead of going to calculus class, im logging onto youtube from now on!
@McGeekSpeak thank ya very much! glad u like the vids.
Thank you for yet another brilliantly explained, concise video. So well explained - I had never done partial fractions before and it made everything so simple. Keep up the good work!!
@29edaj no, i do not have any of that stuff
he said dp hehehe
It took me three watches to notice that you're left-handed
i did not take calc til college. mainly i skipped school a lot during high school instead. it was worthless.
@Taowhr no, i do not! but if P = K, the dP/dt = 0 , which makes sense. the population has reached the carrying capacity, so the change in population would be zero
Hey man, thanks for the video! I know its old but could you explain why you remove the absolute and then say the A can be + or -?
Because the absolute value function always outputs a positive number (the distance of a value from zero), the input of (K-P)/P inside of the absolute value bars can either be positive or negative of what the function outputs for real numbers. Hope this helps!
Thank you for this simple explanation! Unfortunately textbooks fail to understand that if they explained things like you just have - that it would build our confidence and help get our minds around more complex problems.
@29edaj send that big donation when you start making all that $
I'm momentarily confused over where the dp at 4:38 came from ;-;
you can try, but i don't think so.
How can this be used to forecast covid cases?
thanks, im taking dynamical systems right now this is so much help, thanks Patrick
He did that to help with separating P from the equation at a later stage. But an answer can be obtained by various ways.
I tried solving for P(t) WITHOUT multiplying by (-1) and initially got this: [P/(K - P) = Ae^kt] --> Now in order to completely separate P, I'd have to take the inverse of both sides of this equation: [(K-P)/P = A.e^-kt] --> [(K/P) - 1 = A.e^-kt] --> P = K/(A.e^-kt+1), which is Patrick's solution. Hope that clears things out (:
thanx man, you're practically taking me through my 1st year of my actuarial studies
so did u ever become an actuary
The BIG K should replace to "M", so we would not mix up.
I know right in the book it said M as max population
replace small k with r, consistant with almost all representations
MIND BLOWING!!!
Okay I found your channel when I was in algebra 2 and now I’m in calc and you still have clear and helpful videos for that, that’s pretty amazing. Thank you!
Ohhhhh many thanks Patrick this example was a night mare for me and I have skip it as I could not understand it but now I have and also it was one of the last paper exams with 20 mark many thanks for your effort now I can guarantee that I will get it patrik you are amazing xxxxxx
I was making some silly errors which was giving me something totally off. But your video helped me identify where I was making the error and I got the correct answer. Thank you for the video!
thank you so much! this was really helpful :)
you're a hero! I couldn't thank you enough
Just one doubt you took A and B both value equal to one but at different P one for zero and one for K but kept both in single equation how come. Are A and B a constant.
This helps so much thank you!
it's better if you change k to R implying the growth rate since it confuse with your writing between K and k.
0:15 big k and little k you could have utilize different variabale for ease
why do people choose to multiply by negative 1?
Thank U so Much Patrick.......
I swear to GOD, I learned more from just 1 night of studying using UR videos rather that did from years......
Thank U sooooooooooo Much....I wish U could come to South Korea n be my professor of Math in our University ^^
you are totally better than my lecturer lol
thank you Patrick. ^^
Thanks Patrick - I've been struggling with understanding rather than learning parrot fashion - this equation (and the Gompertz) for a month, and the analytic solution has been a big struggling point but YOUR video and YOUR clear voice and step by step explanations of what you are doing seems to have helped. I am not sure your approach is strictly necessary when it comes to the partial fractions (there are probably many ways of doing that) but your twelve second explanation of the "N substitution" part which some books and videos gloss over with a "Oh it's a negative..." here you tells your viewers WHY. SO Thanks. K (Scotland - 57 yrs young...)
Is solving a differential equation is the same as finding the antiderivative? Just like in this question, we know dP/dt, then the solution is to find P(t)=______. Isn't it the same as antiderivative? And if it is, why don't we just use indefinite integration?
Sort of. An antiderivative is a special case of solving differential equations. An antiderivative is the special case where the derivative is a first order derivative, and is directly given, only as a function of the independent variable. The solution (always), is to integrate the given derivative.
A differential equation general, could have a derivative equal to a function of BOTH the independent variable AND the dependent variable. It could also involve higher orders of derivatives, such as a second order differential equation, that is a combination of the dependent variable, the derivative, and the second derivative.
what i can say is AMAYZING! :D
Thankys!
Thanks :-)
Absolutely amazing! I'm an IB student doing my math internal assessment and this literally saved me. Best math video i've ever watched thank you
does anybody know why he doesn't include +c when integrating the left side 4:53? i'm confused about it
Essentially, it's redundant, since he already has a +C on the right hand side. He could write +C1 on the left, and +C2 on the right, which would be a more rigorous way to do it. Then, when he subtracts C2 - C1, that is just a constant on its own, and he assigns C to equal that constant.
You have a similar situation happening at 7:15. When he splits apart the exponential, he turns the constant C into e^C, and e^C itself is a constant. At this point, you can say "reassign C to equal e^C".
Generally speaking, at the end of the solution, there is one arbitrary constant for every order of derivative you have. Even if you generate more arbitrary constants along the way, several of them may be redundant, and can collapse to a single constant.
@jukeboxheroes inverting the fraction (P/K-P) eliminates a potential future division by 0.
at 6:06 why did u multiply the whole thing by -1? ive been doing some exercises and some of them require that i multiply by -1 but some do not. i dont get this at al. ;x
Your videos are absolutely fantastic !! Thank you so much for all the help! Im just wondering if you could please do a video in radical equations and inequalities please :) Thanks soo much !!
Very good video, thank you, but why would you but K and k?? That;s just mean
I always get ln |P / K-P|
why is it important to flip this??? why is it important to multiply by -1???
I like how very clearly you explain what you are doing, this was a lot of help to me.
@patrickJMT do you have anything on propositional logic? .... am in need of that stuff
Thank you soo soo much for the vid!!!
How to calculate K
PATRICK! please tell me WHY you multiplied by -1 at 5:55 !!!
do u know how to do the second derivative of the solution?
why are you allowed to just set t=0 when solving for A ?
Thanks bro
A dot always looks like dot product to me now
and there we have it... /watch?v=nxs0WbkKZCk
Just echoing what other are saying but this is excellent. Thank you.
you are most welcome :)
Simply the best. cheers
Make more differential equations tutorials!
because (K-P)/P = K/P - 1 (better form?)
Thank u so much .👍👍
How old is this Patrick dude o.o
Around in circles.
Great derivation, but representation of the Logistic formula is hard to reconcile with other
generally presented version of the formula.
Interesting name you have. Perfect for someone interested in diffEQ.
@@carultch not a coincidence--
ansatz was taken!!
congrats on 1 mil subs patrick
thanks! :)
Crack!!!!!
Ugh
THANK YOU
THANK YOU
i love u thank u so much
you are very welcome :)
what the value of e
google it