let's now attempt to find a solution for the delicious ticket differential
equation and where he found some constant solutions we could think
through that a little bit just as little bit
love review from the last few videos
so this is the tea axis is this is the an axis we already saw
that it enter 0 at the time equals zero population 0 there's no one to reproduce
in this
this differential equations consistent with that because I any 0
this thing is going to be 0 it sorry to change is going to be easier with
respect to time it so our population just won't change it'll just 80 which is
nice because that's what would actually happen in a real population
so that gives us the Casa solution that and I T 00 that's one solution to this
differential equation not that interesting
zero population will never grow or change the other
concert solution is one of our population stay started at the maximum
move what
the environment could could sustain
in that situation this terms would be k over K which is 11 miles 10
and so on their the population would also not change you just 8k it so the
rate of change would just stay at 0
so that's another concert solution that we start
we start at the max on population and then this difference equation X
expert tells us there that never changes but we said hey look willed it could be
something interesting that happens
if we if our initial conditions she called
and some not butter and o0 yet time because if we start someplace in between
maybe closer
much below are our eventual ceiling I guess
the for a lot more ventral ceiling this thing is going to be a small fraction
this thing is going to be close to one and so the rate of change is gonna be
pretty close to being proportional to N which is going to
and when you were thinking that we can kinda might look something like that is
and increases our raid a change increases
but then as an approach is K this thing is going to approach won this thing is
going to approach 0
it's going to overwhelm this an R-rated changes might is going to
approach the right so we can imagine a scenario where we asymptote
worry asymptote two words K and
for let let's see if we can solve this to find this
this an atty this entity to find the actual
analytic expression for this NFT read your to this is interesting this is
what could could be used to model populations that would be
more consistent with the male 20 in the mindset
we'll see if we can do that and to do that we just realize this is ade
completely different this is a separable differential equation and were swimming
is a function of teargas all Ford
and up t that satisfies is as we said the separate from the
explicit tease but there are no explicit easier so it's quite easy to do
so what I'm gonna do is undertaking this part
right over here I'm a divide both sides
I'm gonna leave the are on the right hand side it'll make things I think a
little bit
easier as we as we try to solve for entities let me just do that
to this is going to be equal to you one over
an hi games
one minus and over K
one minus an over K
times DND T times
D and d T
is equal to our is
equal to are and
another we could have we could think about their do it. let me just let me
just continue to tackle it this way so we get
we get that an hour wanna do is take the integer have both sides with respect
to T well this is pretty straightforward that is going to be
our tee time some constant what's this going to be this kind this message thing
and maybe if we can break this out if we can expand this into two
fractions to a partial fraction expansion maybe we can come up with an
expression that it's a little bit easier to find the anti derivative of
and so I'm hoping that I can find it a to be where
a over and
plus be over
one minus and over K is
equal to this business is equal to 1 over
an times one minus and
over cable see if you can find it a to be this is partial fraction expansion
the suit unfamiliar to you encourage you to review that part
on kinda can redo search for partial fraction expansion on Khan Academy
so how do we do it will just add these to right over here
so this if we take the some right over here this is going to be
a times this which is a minus
a over K an plus
be times this pas bien all over
the product at the STU so this is going to be and
times one minus and overcame one will think about it
I just want to buy the numerator denominator love this one by one minus
an minded
one minus an over K so a-tens 11 is that ok is that
and times one mine is that ok is that and multiply the numerator odometer this
both by
an bien and times that and then I added the two now that I had this
the same denominators that's just adding fractions with unlike denominators
it's going to be equal to this one over and times one minus
and over K an auditor they will know what what is
what is 18 be going to be able to work and they be able to
well have a constant term yeah I don't have any interviewer I could say that
maybe have a zero
times in n actually helps things a little bit because maybe we could say
that
this thing this plus this
which are the coefficients on an are going to sum up 20
and that this which is our a
is going to be equal to what that's pretty nice
so we could say that a physical one
a physical one and that if a physical one
we have negative one over K
-1 over K plus be
busby is equal to 0 well what's be going to be
will be. would be equal to you one
one over K so we can rewrite
this as we can rewrite it has
one over an one
over an one over N
plus one over and plus
one over K plus one
over K over
over although all over
we just do this over one minus and over K
one minus and over K
over K and then and then the
hi ames times DND T
the and DT is equal to
is equal to are so I just a little bit a partial fraction expansion
and I give myself a little bit a real estate let me clear this out
you could rewind the video interview that if you find it necessary
so let's do that and so how does this help us well you know I can only pray
might be recognized me into drew have won over and you might even see this was
assisting to use a little bit we know the editor or whatever it is a natural
log
to the new car is the natural log are the absolute value
event we can see that drivel that with respect
an is equal to one
over an referred by the director this with respect to T
derivative with respect to TV the natural log
the absolute value then is equal to
this computer this with respect to an times the derivative
and with respect to T times D and
DT and we could do the same thing over your notice
I have I have an expression
what would be the derivative this expression right every year
it would be -1 over K if I blame the true with respect and
it would be -1 overcame have a positive one over key here I can even make it
negative
I can I can turn this I can clear this other way
instead of having and a positive there I could have the
executive double negative so a
double negative I have achieved the value that a notice to this
is the guerilla this which makes it very nice for you substitution or you might
be already
used to doing this thing in your head and so we know that the derivative
with respect an you write this is a new color
we know that the derivative with respect to an
the the natural log love one minus
and over K once again this comes to the chain rule it's going to be the
derivative
love this with respect and which is negative one over K
times the derivative love this whole thing with respect to this thing which
is
times one over one minus and over K
which is exactly where what we have over here the Powertec through this with
respect to T
so the derivative with respect to T the natural log love one minus
and over K is going to be the river this with respect
and which is that which is what we just found it's going to be this
so copy and pays it's going to be
that it's going to be that times the derivative a van with respect to T
disc straight to the general DN DT
will notice we have a DND TV we can multiply times
each of these things actually wanna missus do that is to make it
clear cuz this is really you know this isn't so much different liquidations but
sometimes
some the calculus that we learned about to long ago
and even frankly so the algebra it's nice to not skip steps if I distribute
this
East whoops wanna copy and paste
so copy
paste so I have that and then I have
then I hikind making copy
and paste that I was distributing
I miss distributing this business right over here
so I get that we clean it up a little bit
so I have that right over here and of course we have this being
equal to you are if I did the anti-terror this with respect to T
one's gonna get I'm just gonna get this
I am just going to get
this as the anti I was gonna get
this and plus or this minus
this minus this right over that
so let's do that was to be entered a router with respect to T to the left
hand side I'm gonna get the natural log
this the entire true that with respected
T is the natural log above the absolute value
than and then minus
the entire ride through this with respect to T
is the natural log the natural log
the absolute value love one minus
Anna over K is equal to or less not forget what states
and we're gonna have some continuity find a pretty general solutions college
C-one
is equal to the entire do this with respect to T is
are times t maybe put some other constant
plus some other constant just like that
and now I'm going to assume that my NFT
meets this this assumption right over here and so I'm going to assume
I'm going to so assuming assuming
us tumor
assume and ft is going to be
less than K a greater than 0 that means that this thing
this and is always going to be positive and if and is always between 0 and K
that means that this thing is always going to be positive
and so that helps me clear things up a little bit we do
this caller that helps me quick get rid of that
helps me get rid of helps me get Rideau
that IQ is dead through some parentheses here
and what I subtract the C-one from both sides
so me that we get rid of it here it
cut and paste so I had the C-one there and I'm
over writing my own work which is me not making it look as clean as possible
restraint idk read it up a little bit and I have an arbitrary constant
heavenly salt for you at madison other consulate is called the some other
constant mom is called general C
so let me just call this let me just call this
see so clear I was gonna call that
see I am just going to call that
see and I could even simplify this a little bit but actually I just realized
in 30 minutes into this video which is longer than I like to do
so let's continue this in the next guy I was getting excited cuz I'm so close I'm
so close
to solving Ford and have tea that satisfies our logistic difference the
question very very exciting