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#help-27
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dim halo
trying to learn about probabilistic reasoning if all outcomes are equally probable. is this correct?
if everything in a set, S, is equally probable, then:
p(F | S) = the size of the intersection between F & S / the size of S, for any set F
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trying to learn about probabilistic reasoning if all outcomes are equally probab...
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dim halo
trying to learn about probabilistic reasoning if all outcomes are equally probable. is this correct?
if everything in a set, S, is equally probable, then:
p(F | S) = the size of the intersection between F & S / the size of S, for any set F
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trying to learn about probabilistic reasoning if all outcomes are equally probab...
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dim halo
<@&286206848099549185>
worthy raft
Yes that is correct
woven radishBOT
NotABot
dim halo
is the condition that the set is the set of all possible outcomes required for this reasoning?
worthy raft
Not really, I was just trying to be fancy then realized I made a minor misunderstanding that made my reasoning wrong so I had to patch it
In a sense P(F | S) is restricting the set of possibilities to S, so that gives us everything needed
Any other questions or clarifications?
dim halo
Any other questions or clarifications?
so would the first derivation you gave (before editing) work?
i forgot how it went but it looked like it made sense when i read it
worthy raft
Pretty much, the problem was I had P(S) = |S|/|U|, but that's only true if everything in U has an equal probability
But since everything in S does have an equal probability P(F | S) = |F inter S|/|S| will hold
dim halo
But since everything in S does have an equal probability P(F | S) = |F inter S|/...
what was the proof for that again? forgot how it goes
worthy raft
I don't really have a strong proof for it, but essentially I just ackknowledge that P(F | S) = P(F inter S | S) and by treating P(_|S) as a probability function with the universe restricted to S, we can use the formula for getting the probability of a set of size n in a universe where everything has equal probability
dim halo
I don't really have a strong proof for it, but essentially I just ackknowledge t...
i dont quite follow. can u be more explicit?
worthy raft
Hmmmmm
dim halo
so in the previous proof replace p(-) with p(-|S)
worthy raft
Pretty much?
It's like, if the set of all possible outcomes is U, then p(- | U) = p(-) right?
Since everything that can happen must be in U
dim halo
yeah
worthy raft
Just like how if everything in U had an equal probability then P(F) = |F|/|U|, we have that since everything in S has an equal probability P(F | S) = P(F inter S | S) = |F inter S|/|S|
Do you get how I'm doing P(F | S) = P(F inter S | S)?
dim halo
Just like how if everything in U had an equal probability then `P(F) = |F|/|U|`,...
ah, so the initial formula should be P(F) = |F inter U|/|U| but its implicitly simplified to |F|/|U|?
worthy raft
Essentially yeah, since F is a subset of U we will have F inter U = F
worthy raft
Okay great!
dim halo
cos P(F | S) = P(F inter S) / P(S)
P(F inter S | S) = P(F inter S inter S) / P(S)
F inter S inter S = F inter S
worthy raft
Yup! Another way of looking at it if you're familiar with the axioms of probability is that
P(F | S) = P((F inter S) union (F remove S) | S) = P(F inter S | S) + P(F remove S | S) = P(F inter S | S) + 0 = P(F inter S | S)
Read another way, anything in F that isn't in S has 0 probability of occurring in P(-|S), so we can just throw them away
dim halo
okay great that makes sense
worthy raft
Awesome 🙂
dim halo
Just like how if everything in U had an equal probability then `P(F) = |F|/|U|`,...
so this part P(F inter S | S) = |F inter S|/|S|
just making sure im on the same page
we're arriving at this in the same way we arrive at p(F) = |F inter U| / |U|, but replace p(-) with p(-|S) to have p(F inter S | S) = |F inter S| / |S|
worthy raft
Yup! Assuming U has the equal probability property of course
dim halo
yeah
and does S also have to have mutually exclusive elements?
for this whole thing to work
ik U has to be mutually exclusive & jointly exhaustive
does that carry over to S?
worthy raft
I'm not sure what you mean by mutually exclusive here, but I believe so
dim halo
as in both outcomes cant obtain
two outcomes are mutually exclusive = they both cant occur
worthy raft
So what would it mean for U to be mutually exclusive?
dim halo
this is where i got that from
worthy raft
Ahhhh
Okay sure, then since S is a subset of U it will also have that probability yeah
worthy raft
property*
dim halo
okay i see
worthy raft
Both yeah
dim halo
collectively exhaustive too?
worthy raft
...Yes, since by definition of P(-|S) we're saying the probability of - happening when we already know something in S is happening
dim halo
right but that doesnt mean something in S actually happens
worthy raft
Yeah okay let me rephrase, if U has both of those properties with the probability function P(-), then S will have those properties with the probability function P(-|S)
worthy raft
No problem! Glad I could help 🙂
dim halo
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drifting granite
Can somebody review my answers for me?
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Can somebody review my answers for me?
drifting granite
I didn’t catch that. They both would be 3 correct?
foggy vault
mhm
drifting granite
Everything else look okay?
foggy vault
mhm
I cant see the graph being talked about in regards to the second half of the first page
but other wise it seems fine
drifting granite
The second half of the first page is the graph shown in the 2nd screenshot
drifting granite
Thank you!!
foggy vault
np
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royal creek
How do I differentiate $\theta(t)(1-\frac{\gamma t}{2})e^{-\frac{\gamma t}{2}}$ where $\theta(t)$ is the Heaviside function
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How do I differentiate $\theta(t)(1-\frac{\gamma t}{2})e^{-\frac{\gamma t}{2}}$ ...
exotic dagger
product rule, along with the fact that the derivative of $\theta(t) = \delta(t)$ where $\delta(t)$ is the Dirac delta distribution
woven radishBOT
lgkoo
exotic dagger
is that the solution?
royal creek
yeah
exotic dagger
that's basically what you got
exotic dagger
after you expand your last expression into delta(t) multipled by the e^(...) + the remaining which is fine
you can then use the sifting property of delta
i.e. $\delta(t) e^{\frac{-\gamma}{2}t} = \delta(t) e^{\frac{-\gamma}{2} \cdot 0} = \delta(t)$
woven radishBOT
lgkoo
royal creek
ah okay so the way it works in an integral
exotic dagger
yeah, sifting property works even without writing it in an integral against some test function
woven radishBOT
Jacob
royal creek
yeah, sifting property works even without writing it in an integral against some...
ah thanks so sifting property when it's multiplied by a function of t and $\delta(ax) = a^{-1}\delta(x)$ when it's multiplied by a constant?
woven radishBOT
Jacob
exotic dagger
first bit yes, second bit not quite
you need to take modulus
i.e. $\delta(ax) = \frac{1}{|a|}\delta(x)$
woven radishBOT
lgkoo
royal creek
i.e. $\delta(ax) = \frac{1}{|a|}\delta(x)$
thanks so much 🙂
exotic dagger
np
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frosty gyro
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frosty gyro
in the solutions it says triangle PAB is similar to PYX
can someone explain why?
Also for b) there's a typo: x= angle APB
wooden wraith
Quadrilateral ABYX is cyclic (all its vertices lie on circle C2)
In a cyclic quadrilateral, opposite angles are supplementary
From that you can show that the triangles are similar
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frosty gyro
yea idk why I was stuck on this for so long
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viral mantle
I have this equation down to (3a^3b^3)(16a^2b^4) I tried foiling but the answer doesnt match with the answer from a calcualtor. In the calculator instead of foiling they just multiply 3*16 and then combine like terms. Wouldnt you have to foil though?
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I have this equation down to (3a^3b^3)(16a^2b^4) I tried foiling but the answer ...
dawn lintel
I have this equation down to (3a^3b^3)(16a^2b^4) I tried foiling but the answer ...
What is foiling
We can just multiply inside the brackets
viral mantle
FOIl, first, outer, inner, last
viral mantle
The answer is this but i just dont know how to get there.
viral mantle
ohhh
dawn lintel
(a^2)(a^3.b)=a^5.b
dawn lintel
Ohh
dawn lintel
i gues you only use it when there is addition in the problem
Yes
dawn lintel
Good
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narrow kite
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narrow kite
do I get the integrate f and g here first and them I can deduct them?
personally id prob split it up in two integrals to begin with and just deduct the results
but given I combjne it jn one integral like they did, i should get the primitives prior to combing f and g right?
foggy vault
i would say split it
narrow kite
ftc?
foggy vault
OH THAT
narrow kite
makes it faster also covers all grounds
foggy vault
yeah
thats what i would do
I got a bit confused by your question
I would say just combine first then integrate
narrow kite
I recall its the other way around, you confident about the order?
foggy vault
you could honestly do either way for int f(x) - g(x)
yeah either way works
let me give you an example
try this
f(x) = x^2 + 2x
g(x) = 3x +1
first try it by combining first then integrating
then second time integrate first then combine
you should get the same answer
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hexed ocean
Can someone please help me with question 3
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Can someone please help me with question 3
hexed ocean
or is that meant to be your work for q3
meant to be for q3
these are the hints i got
but now idk where to go
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hexed ocean
try evaluting the pmf for p=1/2
The combination thing?
eternal tapir
$P(X = x) = \binom{n}{x}(1-p)^{n-x}p^x$
woven radishBOT
chebyshev's infinite pee norm
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hexed ocean
**chebyshev's infinite pee norm**
alr ill try that
hexed ocean
$P(X = x) = \binom{n}{x}(1-p)^{n-x}p^x$
I got like 0.00000006539
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faint zinc
I got like 0.00000006539
Don't use numbers, just use variables.
Remember if p = 0.5 then (1-p) is also 0.5
Which means what about the pmf of k vs n-k?
(hint, what is the formula for the binomial coefficient?)
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hexed ocean
Which means what about the pmf of k vs n-k?
Is it gonna be 0.5 to the power of 1
Wait no
It’s gonna be 0.5 to the power of 60 in this case
Coz it’s played 60 times right
violet wind
No, you're pulling in data from an unrelated problem
hexed ocean
O
hexed ocean
No, you're pulling in data from an unrelated problem
So just 0.5 to power of n then
violet wind
**chebyshev's infinite pee norm**
No, use the binomial distribution pdf
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hexed ocean
No, use the binomial distribution pdf
NCx times 0.5 to the power of n?
violet wind
Yes
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junior fossil
!help
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stone cliff
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stone cliff
I understand the basics but unfamiliar with notation
for i and ii is it just the first and second derivative of r(t)
meager hollow
ye but its l.rl
ye like find r.
and then the i,j,k components
square and take root
stone cliff
Ah got it ty
meager hollow
yw
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opaque talon
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opaque talon
Im so lost with this integral
for b
I used the bounds of -2, 4
and its not returning me the correct answer
wtf
eager lodge
In an integral area is signed (area below x axis is negative) but we want the actual area, not the signed area here
Notice that the integral from -2 to 0 is negative
opaque talon
In an integral area is signed (area below x axis is negative) but we want the ac...
ah shit i forgot
so I take the modulus
of bounds from -2,0?
and then add to the other area
eager lodge
Yep
digital mulch
whats the difference between BPD AND BCD
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whats the difference between BPD AND BCD
digital mulch
the button on the calculator for binomial hypothesis testing
when do we use which one
sharp root
Mine always say binomial pdf and cdf
So I assume your BPD is binomial pdf and BCD is cdf
digital mulch
yeah when do you know which one to use
sharp root
So pdf is the height of a function at a certain point
Cdf is the area under the curve of the function over a range
You use PDF when you want the probability something happened exactly x times
Cdf for when you're looking at a range of x values
digital mulch
okay thank you
sharp root
For example, a finding the probability a fair die rolled a 6 4 times in a row is PDF
Finding if the fair die is actually fair is CDF
safe jasper
yes
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digital mulch
hey
digital mulch
how do you find what number you test this on
eg. how do you get p(x<=2)
how do you get that number
sharp root
Bc the coin got heads twice
sharp root
what about for this one
What's your question?
digital mulch
heres the question
sharp root
heres the question
But what are you asking me?
sharp root
Bc 7 ppl order a starter
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steep fjord
hello people. i ran some one way ANOVA tests but im unsure how to represents the results i got. is there a table i could use? or a graph?
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hello people. i ran some one way ANOVA tests but im unsure how to represents the...
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steep fjord
no
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crystal sleet
Does this problem have any construction? I have tried the case with 4 columns and 1,1,3,3/1,2,3,4 works. I think there should also be a solution for 2024, but still does not find one.
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prime narwhal
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prime narwhal
Where do I start (I haven't done any questions of this exact type before, and was not taught how).
quaint citrus
Maybe try isolating ‘y’
There’s a trick to these, but if u haven’t done them, that’s the best path
prime narwhal
Maybe try isolating ‘y’
why is that? (just wondering)
i know the formula is y=mx+b or something
prime narwhal
U wanna get it in that form
so get y alone?
quaint citrus
Yes
quaint citrus
Keep going
prime narwhal
Keep going
then would i take away 5x? or how does that work
i dont understand how i get rid of the 5x
do i just d ivide by 5 and leave the x?
quaint citrus
Well it’s a term being added to the left side
So what should u do to both sides, to remove 5x from the left side
prime narwhal
Well it’s a term being added to the left side
So what should u do to both sides...
take away 5x from both sides?
idk
quaint citrus
Wdym by take away
prime narwhal
idk dude
quaint citrus
What does that mean
prime narwhal
sorry
quaint citrus
Yea
prime narwhal
so do i dived by 4?
quaint citrus
Yea
prime narwhal
y=-1.25x+20
quaint citrus
Yea
quaint citrus
Yea
quaint citrus
It’s in the form y = mx+b, what do u think
quaint citrus
Ur correct
prime narwhal
Ur correct
how do i figure out this one
quaint citrus
Plug in x = 0 and solve for y
Plug in y = 0 and solve for x
prime narwhal
Plug in x = 0 and solve for y
Plug in y = 0 and solve for x
6x+5(0)−90=0
so do this
and i will find the x intercept?
quaint citrus
Yes
weary hollow
Sorry for interrupting but would it not be 6x+5y=90
prime narwhal
Sorry for interrupting but would it not be 6x+5y=90
no problem, what did i do wrong?
weary hollow
Well at least how I was taught you add 90 to the -90 so you can move it to the right of the equal sign
Then split the equation
quaint citrus
6x+5(0)=90
What will u do after this
weary hollow
6x=90 and 5y=90
prime narwhal
Well at least how I was taught you add 90 to the -90 so you can move it to the r...
i thought i would add 90 to both sides getting rid of -90 on the left and adding 90 to 0 on the right which is 90?
is that wrong?
prime narwhal
What will u do after this
subtract 5?
6x=85
uhh i think i did something wrong
where did i make a mistake
prime narwhal
but y is 0 since im finding x intercept
weary hollow
Oh your doing it that way
I split the equation into 2 equation
6x=90 and 5y=90
Then divide 6 to 90 and 5 to 90
quaint citrus
That’s the same thing, u just skipped some steps
weary hollow
Correct
weary hollow
Correct 👍
prime narwhal
weary hollow
Now find “y”
prime narwhal
Now find “y”
i don't need to since its multiple choice
and thats the only option with 15, 0
weary hollow
It’s better because there’s some options were x is 15 on other choices
It’s easier now but later on it will screw with you
quaint citrus
and thats the only option with 15, 0
I mean I think this is good logic
It’s the only correct x - intercept value
Sure if u wanna practice go ahead, but it’s not necessary at all
weary hollow
It’s just better in my opinion to work for the other answer
But yeah I agree with Stephen
quaint citrus
Yea to make sure 100%, I know what u mean
weary hollow
Alr Stephen sorry for interrupting
quaint citrus
No worries
prime narwhal
Alr Stephen sorry for interrupting
you still helped so all good 👍
quaint citrus
There are lots of great videos on the internet for this kind of stuff, just search “how to find equation of line with 2 points”, the resources are great
I don’t think u really need me with that stuff out there
prime narwhal
There are lots of great videos on the internet for this kind of stuff, just sear...
i try that stuff but i often strugle, its easier when i talk with someone with the exact question im doing so they can make sure im on track
i understand if you can't or don't want to help me but its much easier for me
quaint citrus
Yea honestly these problems are kind of boring sorry
orchid hearth
If I understand this question correctly, you are supposed to find the equation that goes trough the two points, right?
prime narwhal
If I understand this question correctly, you are supposed to find the equation t...
i belive so yes
i still dont understand what general form is but
orchid hearth
right!
prime narwhal
77 - -67 / 12 - -12
144/24
72/12
36/6
6/1
so my slope is 6/1?
which is also 6 (we we are at y=6x+b)
prime narwhal
do you already know how?
is b the y intercept?
orchid hearth
english is not my main language but yeah b is the point where x=0
so, do you know how to calculate b in our example?
orchid hearth
you insert a point into your general form, meaning you already know the slope and u also insert one of the points given in the task, then u solve for b
so y=6x+b
and then u insert either (77 12) or (-67 -12)
and then solve for b
prime narwhal
and then solve for b
i thought we need to set somethign to 0
orchid hearth
well at the end u have the y coordinate for x=0
prime narwhal
well at the end u have the y coordinate for x=0
what do you mean
im confused at this step
orchid hearth
i thought we need to set somethign to 0
i was just trying to explain what b is with this message, not how to calculate it
just insert a point from the task into y=6x+b
prime narwhal
just insert a point from the task into y=6x+b
so how do i insert 12, 77
orchid hearth
well we have 12=x and y=77 so we put those number into they're respective places so we end up at: 77=6*12+b
prime narwhal
well we have 12=x and y=77 so we put those number into they're respective places...
👍
orchid hearth
so know we solve for b
prime narwhal
so know we solve for b
so get b alone?
orchid hearth
yes that is right
orchid hearth
yes
orchid hearth
yep
orchid hearth
thats a great question, afaik that is not the general form but that doesnt quite matter just check which one of these terms is equivalent to our term
prime narwhal
thats a great question, afaik that is not the general form but that doesnt quite...
d.?
prime narwhal
good job, jus lemme know if you have more questions
yes i have two more math problems
orchid hearth
shoot
prime narwhal
shoot
im not sure how i do this
Convert 10x+y+5=0
to slope-intercept form, and choose the graph that represents it.
orchid hearth
okay now i get what your previous question was hinting at with general form
here you need to convert the term into the y=mx+b which your task calls slope-inercept form, and then you can check what graph fits best
prime narwhal
here you need to convert the term into the y=mx+b which your task calls slope-in...
so to convert it i need to get y alone?
orchid hearth
yeah
prime narwhal
- it goes down?
orchid hearth
yeah
prime narwhal
- -5 is the y intercept?
orchid hearth
1. it goes down?
by what factor?
orchid hearth
2. -5 is the y intercept?
yes
prime narwhal
by what factor?
by -10
orchid hearth
yes
prime narwhal
so is it this one?
orchid hearth
thats correct
prime narwhal
thats correct
one more question
prime narwhal
thats correct
orchid hearth
well this looks quite the same doesnt it
prime narwhal
well this looks quite the same doesnt it
yes it does
so lets get y alone
4x−y+12=0
−y=-4x-12
orchid hearth
no, not quite
prime narwhal
no, not quite
where did i make a mistake?
orchid hearth
you can just add y to the originial term
prime narwhal
what do you mean by that
orchid hearth
you got the prefix (i hope hats what it is called in english, i mean the -) wrong
prime narwhal
ohh
orchid hearth
but you then just turned the prefix from the y
prime narwhal
you got the prefix (i hope hats what it is called in english, i mean the -) wron...
whats your main language?
orchid hearth
german
orchid hearth
yea
prime narwhal
i have a few german friends
orchid hearth
well you had the correct term: -y=-4x-12
and then u just put: y=-4x-12
which you cant do
it would be y=4x+12
orchid hearth
yessss
prime narwhal
since -y is just like -1y
prime narwhal
now we have the tern which graph fits this term?
y=4x+12
orchid hearth
yes
orchid hearth
yea while your answer is correct u should probably prove it by looking at the slope not at the y intercept
espacially when the y intercept is not on the screen or picture and multiple lines could get there
prime narwhal
yea while your answer is correct u should probably prove it by looking at the s...
whats the best way to use that to prove it
orchid hearth
look at two points you can easily determine like (-4 -4) and (-2 4)
orchid hearth
77 - -67 / 12 - -12
and then you devide the difference between the two y´s through the difference from the two x´s just like up here
orchid hearth
jop
prime narwhal
so grab like a y2 a y1 a x2 and a x1
i won't do it now just becuase i know but
just for future
orchid hearth
yes
prime narwhal
(this one i dont need to hand in work its a practice test)
prime narwhal
yes
okay tysm, i should work on some other classes now but would you maybe be able to help me in the future
orchid hearth
yeah sure just dm me or smth
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solemn sinew
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safe fractal
What kind of backwards graph drawing shenanigans is this?
solemn sinew
,rotate
woven radishBOT
safe fractal
Your horizontal asymptote is wrong
Your vertical asymptote is correct
you drew it wrong though
Also you factored out wrong in the top part of the fraction
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signal lava
!help
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signal lava
hi
Ben is 20 years older than Daniel. Ben and Daniel first met two years ago. Three years ago, Ben was 3 times as old as Daniel How old is Ben now?
the first sentence can be represented as b = d + 20
the last sentence can be represented as
b - 3 = 3(d - 3)
if we plug in b into the equation you get
d + 20 - 3 = 3(d - 3)
but notice there are no b variables how are we suppose to solve for b then
frozen aurora
first solve for d
frozen aurora
then plug into b = d + 20
signal lava
oh ok
frozen aurora
an easier way would be to do
d = b - 20
so you get b-3=3*(b-20-3)
you solve for b and that'll be your answer
signal lava
hang on
ill be right back give me like a minute
@glacial nimbus
@frozen aurora
is d = b - 20 the same as d = b + 20
oh sorry
wait nvm it is
frozen aurora
is d = b - 20 the same as d = b + 20
no
frozen aurora
you are confusing b and d lol
signal lava
is d = b - 20 the same as b = d + 20
frozen aurora
$b = d + 20 \iff d = b -20$
woven radishBOT
artemetra
frozen aurora
no problem!
signal lava
so
would it look like this
b - 3 = 3(b - 20 - 3)
or am i missing a parantheses
or like b - 3 = 3((b - 20) - 3)
what
how did they simplify this
isnt it 3(b) 3(-20) 3(-3)
oh
wait
nvm
well
i-3 * -3
wait
oh nvm
fairyose
i keep miswriting the numbers
like b - 3 = 3b - 60 + 9
maybe i have trouble seeing numbers
and negative signs
its suppose to be a -9 not a positive 9
how can i prevent silly mistakes like this
it happens to me so often
hmm ok
sure
devout snowBOT
@signal lava Has your question been resolved?
ornate steeple
I need to find the continuity, derivability, differentiability of f(x,y), I tried with y=mx and parametric function (t,t^2) but I actually dont know what to do
devout snowBOT
I need to find the continuity, derivability, differentiability of f(x,y), I trie...
teal perch
What is sen(x^2)
dense lynx
sin
teal perch
Hmm try other functions other than linear and parabolic ones
Maybe model one
Such that it somehow cancels out
ornate steeple
How can I model one?
devout snowBOT
@ornate steeple Has your question been resolved?
devout snowBOT
@ornate steeple Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
grizzled roost
Let $\mathbb{L}^d=(\mathbb{Z}^d, E)$ be a graph where an edge ${x,y}$ is in $E$ if $d(x,y)=1$. Let $(\omega e){e\in E}$ be i.i.d. Bernoulli random variables of parameter $p$. We say an edge $e$ is open if $\omega _e =1$. For a given $x$ in $\mathbb{Z}^d$, let $C_x$ be the set of vertices that are connected to $x$ via a path that has only open edges. Let $\theta (p)$ be defined as $\mathbb{P} (|C_0|=\infty)$.\
My question is: how do we know that $|C_x|$ is a random variable for each $x$? And how do we know that the probability on that is well defined? That is, let there be two different probability spaces $A_1$ and $A_2$, both satisfying everything that is said above. How can I know that, $\mathbb{P}_1 (|C_0|=\infty) = \mathbb{P}_2 (|C_0|=\infty)$ (for a given $p$)?\
The reason behind this question is that there is a proof that relies on adding structure to percolation. It begins by defining $U_e$ uniform iid random variables, such that $\omega_e = 1$ iff $U_e < p$. Those weren't there at the beggining so it is not obvious that this new object is actually the same percolation. The same can be said for many other proofs, that basically study one particular case of percolation model (with more structure) and then claims that the result is also true for all percolation models.
devout snowBOT
Let $\mathbb{L}^d=(\mathbb{Z}^d, E)$ be a graph where an edge ${x,y}$ is in $E$ ...
woven radishBOT
Casiel368
devout snowBOT
@grizzled roost Has your question been resolved?
polar flax
devout snowBOT
polar flax
I do not understand how they were able to get the answer
especially with the last few steps
The rest of the steps on the next page
But I need help with this
raven leaf
Be specific
polar flax
Okay
So like in the last few steps
How did they go from d(u(t)y/dt to d(u(t)/dt
That is one of the weird parts in this question
raven leaf
Product rule
polar flax
I understand that part
But how do they go to the step after that
And remove the y from the differential equation
tall knoll
this part seems to be awkwardly typeset
tall knoll
that last term should read $\frac{\dd \mu(t)}{\dd t} y$
woven radishBOT
Steakanator
raven leaf
Product rule
tall knoll
both mu and y are functions of t
tall knoll
so as mentioned, this is just the product rule
tall knoll
this?
tall knoll
this part seems to be awkwardly typeset
compare this with the second equation in your screenshot
both have a dy/dt*mu(t) term
the other term in each must be equal
polar flax
I am sorry can you type out what you mean
Also, why is there an "m" the one you wrote
woven radishBOT
Steakanator
vapid nest
mobius function
tall knoll
$\text{second equation: } 4\mu(t) = \mu(t) \dv{y}{t} + 2\mu(t)y \ \text{third equation: } \dv{\mu(t)y}{t} = \mu(t) \dv{y}{t} + \dv{\mu(t)}{t} y$
woven radishBOT
Steakanator
tall knoll
compare the right hand side of each equation
polar flax
Okay
So the u(t)(dy/dt) cancel out
So we have
4(mu(t))-d(mu(t)y/dt = 2(mu(t))y-(d(mu(t))/dt)y
Right?
tall knoll
i'm not sure why you're subtracting things
polar flax
Oh wait
I'm an idiot
Oops
Ummmm
Can we subtract the d(mu(t)y/dt from both sides of the bottom equation
tall knoll
if you want
polar flax
What am I supposed to do
tall knoll
re-read what they've written after the second equation
polar flax
I did, multiple times
tall knoll
their argument is that the lefthand side of the equation is the derivative of some product of functions, ie
$\dv{\mu(t)y}{t} = 4\mu(t) \text{ for some function } \mu$
polar flax
Yeah
woven radishBOT
Steakanator
tall knoll
you have an equation for 4mu(t) (which is precisely the second equation)
polar flax
Your equation only had the 4(u(t))
So, because there was a 2(u(t))y
Added to it
I subtracted it onto the other side
Is that wrong?
tall knoll
it is wrong
polar flax
How
tall knoll
So, because there was a 2(u(t))y
Added to it
which equation are you referencing?
polar flax
The second one
tall knoll
you're conflating $\dv{\mu(t)y}{t}$ with $\mu(t) \dv{y}{t}$
woven radishBOT
Steakanator
polar flax
The way they used it in the book I thought they were the same
tall knoll
the first is the derivative of the function "mu*y"
the second is the function mu multiplied by the derivative of y
polar flax
Oh okay
Cool
Can you tell me how they were able to compare the 2nd and 3rd equations to get the 4th equation
tall knoll
after the second equation, their argument is that the lefthand side looks like it's the derivative of some product of functions, ie
$\dv{\left[\mu(t)y \right]}{t} = \mu(t) \dv{y}{t} + 2\mu(t)y$
clear?
woven radishBOT
Steakanator
polar flax
Ummm
tall knoll
i'll tack on some brackets to make it as clear as possible
polar flax
But they wrote something different in the book?
tall knoll
ignore the book for a second
polar flax
But how do I know how these are related then
tall knoll
the lefthand side looks like it's the derivative of some product of functions,
the equation I just posted is what this argument means
tall knoll
their exact words: "The left side of the previous equation [second equation] looks very much like differentiating the product mu(t)y"
tall knoll
**Steakanator**
which directly translates to this
polar flax
Okay
tall knoll
if you have any doubts about this, ask them now before we move on
polar flax
Umm okay
How do we get the 2(mu(t))y
In the equation, because they don't mention it in the sentence
tall knoll
that's part of "the left side of the previous equation"
polar flax
So does that mean that they are implying d(mu(t))y/dt = 4(mu(t))
tall knoll
that's the next step yes
tall knoll
**Steakanator**
the righthand side of this is the same as the lefthand side of equation 2
so therefore the two other things must be equal as well
polar flax
Yes
tall knoll
the next step is to determine what precisely mu is
polar flax
Okay
tall knoll
for any mu, we can calculate the derivative of mu*t using the product rule
polar flax
Yeah
tall knoll
**Steakanator**
we also know that the derivative of mu*t must follow this
so we know that:
polar flax
Yes
tall knoll
$\dv{\left[\mu(t)y \right]}{t} = \mu(t) \dv{y}{t} + 2\mu(t)y \ \dv{\left[\mu(t)y \right]}{t} = \mu(t) \dv{y}{t} + y\dv{\mu(t)}{t}$
woven radishBOT
Steakanator
tall knoll
now we compare terms
polar flax
Okay
tall knoll
the lefthand sides are the same, the first term on the righthand sides are the same...
what about those last terms?
tall knoll
right
tall knoll
we can
polar flax
Okay
2(mu(t)) = d(mu(t))/dt
So we have
1/(mu(t)) d(mu(t)) = 2 dt
Is that right?
tall knoll
seems so
tall knoll
your lefthand side should be ln(|mu(t)|)
tall knoll
there's no guarantee that mu is positive yet (which would fall outside of the domain of ln if broken)
polar flax
Yeah
Okay
ln(|mu(t)|) = 2t + C
So we have
|mu(t)| = e^(2t+C)
Which is
|mu(t)| = Ce^2t
Is that the right answer?
tall knoll
you should resolve the absolute value situation first
woven radishBOT
Steakanator
polar flax
$|/mu(t)| = \begin{cases} t & t>0 \ -t & x \leq 0 \end{cases}$
That might be wrong
Maybe
woven radishBOT
Player
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tall knoll
$|\mu(t)| = \begin{cases} t & t>0 \ -t & t \leq 0 \end{cases}$
woven radishBOT
Steakanator
polar flax
Oh thanks
tall knoll
this?
tall knoll
ok no, that's wrong
tall knoll
you're looking at |mu(t)| not |t|
polar flax
I think I know
Yeah
I should have known that
$|\mu(t)| = \begin{cases} \mu(t) & \mu(t)>0 \ -(\mu(t)) & \mu(t) \leq 0 \end{cases}$
woven radishBOT
Player
polar flax
Is it that
tall knoll
we're getting closer
polar flax
Hmmmmmm
tall knoll
|mu(t)| = Ce^2t
here you had mu on one side and some exponential on the other
try to do that with your current equation as well
polar flax
So do I have to turn it into a piecewise function?
tall knoll
you already have one
just sub in the exponential on the righthand side of your current piecewise
tall knoll
because our ultimate goal is to sub in an explicit function of t into the original diff eq
polar flax
Oh okay
tall knoll
mu(t) is just some unknown function; e^t and its cousins are things we can actually integrate and such
polar flax
But we know what mu(t) is equal to
polar flax
Oh okay
$|\mu(t)| = \begin{cases} \mu(t) & Ce^(2t)>0 \ -(\mu(t)) & Ce^(2t) \leq 0 \end{cases}$
woven radishBOT
Player
tall knoll
botched tex aside, that's getting closer
polar flax
Good
tall knoll
$|\mu(t)| = \begin{cases} Ce^{2t} & Ce^{2t}>0 \ -Ce^{2t} & Ce^{2t} \leq 0 \end{cases}$
woven radishBOT
Steakanator
tall knoll
this is what i was hoping you'd get
tall knoll
now, the next important question: what is -C?
tall knoll
i mean in general
polar flax
Hmmm
tall knoll
if C is some constant, what can you say about -C?
tall knoll
that's unfortunate
polar flax
Hmmm
tall knoll
ln(|mu(t)|) = 2t + C
think back to this
tall knoll
waiting
tall knoll
we're not replacing anything
tall knoll
ln(|mu(t)|) = 2t + C
you integrated both sides, why is there only a constant on one side?
polar flax
The +C is in the exponent
tall knoll
there is no exponent
polar flax
Oh
I though you were refffering ot the the one I just sent
For that we can move all expoenetns to one side
tall knoll
1/(mu(t)) d(mu(t)) = 2 dt
you integrated both sides of this, but there's only a +C on one side. Why are you allowed to do that?
polar flax
Because we would have 2 constant on both sides, then we can subtract one of them onto the other side and rename it as just C
tall knoll
so a constant plus (or minus) another constant is itself just a constant
tall knoll
so if we have some constant C, what can we say about -C?
tall knoll
it is just another constant
polar flax
That took me way too long.
tall knoll
so we're thus able to write $\mu(t) = Ce^{2t}$ for some carefully chosen constant C
woven radishBOT
Steakanator
tall knoll
and we're golden
polar flax
Wait
But how to we know C=-C, I thought we would only be allowed to replace -C with something like D
tall knoll
i tend to name all my constants the same thing since they're arbitrary anyway, but we can if you want
$\mu(t) = De^{2t}$
woven radishBOT
Steakanator
