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Math 219 Lecture 1 — Transcript

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1:18foreign
1:41foreign
1:44welcome everyone
1:47um this is 219
1:49differential equation course
1:52I meet you all but if you want to talk
1:55you can unmute yourself
1:58and ask your questions
2:00well this is our syllabus let's go over
2:03the syllabus first
2:05um
2:18thank you
2:24all right well
2:33here is my name uh usually I use gem
2:37um in the park I am a member at the math
2:40department
2:42guilty member and our course assistant
2:45is and his office is at as if and my
2:50office is at
2:54um
3:01m217
3:04217 at Math building
3:15one two seven and office hours are
3:19um
3:25Tuesday
3:30uh 2 30
3:33to 4 30.
3:35so during this time I will be in my
3:38office you can come and ask your
3:40questions you can also
3:43um
3:44attend office hours to zoom uh link I
3:49will also open the zoom link during my
3:51office hours
3:53so if you have any question
3:55about the course you can come and ask
3:58your questions okay
4:06all right
4:18well here is the website of the course
4:22we make some announcements through this
4:25website but from now on you should
4:28follow what the class for the
4:30announcements all right we will do most
4:33of the announcements to Audi class here
4:36okay you can find the updated syllabus
4:39there
4:44and this is our textbook voice and the
4:47Prima
4:48um
4:49it's good to have one to do some
4:53exercises
4:55well
4:57on the course website there are
4:59suggested exercises so you can go ahead
5:03and
5:04try to lose problems
5:06from the book
5:08as for the attendance I will not take
5:11any attendance but of course you should
5:14follow the course you should come to
5:17each lecture okay but this is not a must
5:20I I would say
5:22no attendance will be taken on the
5:24course but
5:26um you should definitely come to the
5:28lectures to get a good grade
5:31uh uh well let's talk about the grading
5:33we have two midterms and one final exam
5:36so we're planning to do this mid terms
5:39in class in class midterms
5:42um well these are the dates
5:45uh 30 percent each midterm and we have
5:4940 percent for the final total it is 100
5:52percent no quizzes no recitations this
5:56is only four hours course okay
5:59um so we will meet on Tuesday and Friday
6:02right
6:07all right
6:19so
6:22well let's
6:28go to our web page here
6:40you can Google it
6:43for the web page
6:47it's not this one
6:58by the way I'm taking recording and this
7:02I will put on autoclass after each
7:05lecture you will see you will able to
7:07see
7:08um
7:09the lecture video recordings and
7:12I will also put the collection notes on
7:16a C-Class
7:18please
7:21all right so this is our
7:23[Music]
7:23um
7:24web page
7:26well there is here for instance you can
7:29find suggested problems
7:33a number here the 490 Edition and 10th
7:37edition there is also 10th edition of
7:39this book okay
7:41so you can check this so these are
7:44important because uh this will help in
7:47the exams okay there are many
7:50exercises here
7:55uh well there is one question am I going
7:58to upload the video on YouTube
8:02um
8:03I guess it will automatically upload it
8:06to YouTube and also I will put a link
8:10um to the Audi class so you can also go
8:13ahead
8:14and watch from multi-class there will be
8:17a zoom link okay and Dot class for the
8:20videos
8:26all right well
8:31there is here post past exam questions
8:34you can see some examples of the last
8:38year's exams
8:42here you can see the syllabus
8:45foreign
8:53syllabus maybe it's better to go with
8:57um
9:00it doesn't open well I guess it's about
9:04the iPad because I did this earlier what
9:09uh
9:10with the computer anyway you can also
9:13find the syllabus on articles
9:17all right what else we have here
9:23yeah that's it I guess yeah
9:29well
9:31our makeup policy is univ will need an
9:34acceptable excuse like a report medical
9:37report or something similar to take a
9:40makeup exam I don't suggest makeup exam
9:43but if you have to take it
9:45then you can take it with with some
9:47report
9:49you can only take one makeup exam be
9:51careful with that okay so you cannot
9:53take two makeup exam for two midterms or
9:57one midterm one final it's not a lot
9:59okay so you can take only one
10:02and
10:03this makeup exam will be after the final
10:06we will do announcements for this
10:12here is DNA policy if you miss uh two
10:17midterms you will get an A if you miss
10:19uh one midterm and your other midterm is
10:23less than 15 over 100 then you will get
10:26an A again
10:27if you take two midterms again two
10:29midterms and your total is less than 30
10:33points then you will get okay so this is
10:35clear in the syllabus
10:47all right
10:53here we have the links for the zoom
10:56meeting and I guess the updated syllabus
10:59has also YouTube links
11:02this is the older version
11:09and we just talk about office hours
11:12office hours are on again there on
11:16Tuesday
11:18uh from 2 30 to 4 30 Tuesday 2 30 to 4
11:2430. let's go here
11:27um
11:28not this one
11:32well
11:34here is the course schedule we have a
11:36detailed schedule for each week we will
11:38follow the schedule so for instance
11:41today
11:42um we're gonna do one one two two and
11:46two one okay
11:48uh sorry one one and two two and on
11:52Friday we will do 2.1 linear equation
11:56integrating factors and so on
12:04so in this course we will learn how to
12:07solve differential equations of course
12:09we cannot solve all types of
12:11differential equations we can only solve
12:14in certain type differential equations
12:17and we will focus on this okay
12:25all right
12:26um before we start do you have any
12:28question
12:39hmm okay there's one question some of
12:42the suggested problems do not have the
12:44solution key is there a way to learn
12:45those well
12:48um I I don't I don't have any
12:50information about that
12:53um what you can do is if there is no
12:55solution key you can come and ask me in
12:58during the office hours you can do that
13:02do you have a suggested way to
13:05um
13:07uh study this course
13:10suggested what
13:12suggested way to study this well
13:15suggested ways like um try to solve this
13:19uh exercises from the book
13:22that will help a lot because
13:24differential equation is about solving
13:27differential equations so you need to
13:29practice more and more
13:31and we have here suggested problems on
13:35the web page you can go ahead and try to
13:38solve after each lecture try to solve
13:40these problems this will help a lot okay
13:42so because in the exams you will you
13:45will see similar questions so that will
13:47help then that will be enough
13:51foreign
13:55and of course if you cannot solve the
13:58problems here you can ask me you can
14:00come to my office hours you can go to
14:03your course assistant
14:05and ask him okay
14:16all right any other question any other
14:19comment
14:25well during the lecture you can stop me
14:28and ask any questions you want okay
14:30anytime you can
14:32unmute yourself and ask your questions
14:38okay
14:41now let's start with the basics of
14:44differential equations
15:14now this is mod mats
15:19219
15:28spring semester
15:42all right
15:46[Music]
15:48right well sometimes people Call it
15:51spring 2000.
15:5322 and 23
15:56so let's catch
16:07all right
16:09um
16:10let's start with the introduction
16:20questions
16:22so we will Define what a differential
16:26equation is
16:28this is 1.1 in the book
16:31okay the definition
16:37what it is a differential equation well
16:41you already took calculus one and two
16:45and in the calculus
16:46um most of the time you are given a
16:49function and you do some work with the
16:51function you find a derivative you
16:53integrate
16:55or you are given a curve and you find
16:57ARIA blah blah right so we are given
17:00explicitly something and you do
17:01something with that given function
17:05in this course we will do kind of the
17:07reverse so we are not given the function
17:10we are given an equation which is called
17:13the differential equation and we will
17:15try to find
17:17uh the function which defines this
17:20equation okay
17:22so a differential equation is this
17:27a differential
17:38X
17:39equation
17:44just
17:47an equation involving one or more
17:50derivatives
17:52yes
17:54an equation
17:58annual way
18:03one or
18:06more
18:08derivatives
18:14of
18:15an unknown function
18:25okay so unknown function is like we will
18:29call it like y
18:30and it's a function of x t whatever you
18:34call it
18:35okay
18:39well this is our first definition so
18:42let's we will do some examples later
18:45let's give this definitions first the
18:48highest derivative in the differential
18:50equation is called order of the
18:52differential equation the highest
18:58foreign
19:05in a differential equation
19:15yes
19:18called
19:22the orders
19:28of the differential equation
19:44okay
19:48another definition so let's say you have
19:52for instance in the differential
19:54equation you have the highest order is
19:57like three you have
19:59derivative of order 3 and the
20:02differential equation now you say this
20:04differential equations of order three
20:07just
20:11and a differential equation is called
20:14Ordinary differential equation if there
20:16is only one independent variable
20:19at the thresh kill
20:22equation
20:27is called
20:31and
20:32ordinary differential
20:45ordinary differential equation
20:51okay
20:53for short we will write ode okay
20:57or the
20:59if
21:04there is only one
21:10one independent
21:15variable
21:17that means what independent variable is
21:19like
21:20for instance
21:23you have let's say Y is equal to Y of x
21:28okay
21:29it's a function y is called dependent
21:33variable X is called independent
21:35variable okay
21:38Y is dependent
21:44variable and there is only one
21:47independent variable here
21:50which is X x is
21:55the
21:57only one
22:01independent
22:09variable well if you have for instance Y
22:12is equal to y
22:15x t
22:17then there are now two variables two
22:20independent variables x and t
22:24X
22:25t
22:26R 2 independent
22:30of course these are not differential
22:31equations these are simply the functions
22:34okay so here we Define what's
22:36independent and independent variable
22:41X tr2
22:43independent
22:46variables
22:49okay
22:54so if there is only one dependent
22:56variable then you say it's an ode
22:59ordinary differential equations if there
23:02are more than one
23:05independent variables then you see it's
23:07a partial differential equations let's
23:11also write this
23:13if
23:16there are
23:19more than
23:25one independent
23:32variables
23:37then
23:40differential equation is called
23:48foreign
23:52partial differential equations
23:57partial differential
24:03equation
24:05quick question
24:07is in this case we write pde for short
24:13well this partial comes from partial
24:15derivatives
24:16remember if you have two variables in
24:19calculus
24:20then the derivative you can take to
24:22derivative with respect to X and Y and
24:25these are called partial derivatives
24:27right so if your equation is in terms of
24:30partial derivatives then you say it's a
24:32partial differential equations PD
24:37any question so far
24:41um so let's do some examples
24:44uh
24:45I
24:47this is going to be a weird question but
24:50is there is there any sort of gray gray
24:54area between
24:55Odes and P well let's start with the
24:58simple one
25:01d y d t that means the derivative of y
25:05with respect to T So Y is a function of
25:08T is equal to Y minus sine of t
25:19okay there's
25:21huh
25:23well I couldn't hear you can you ask me
25:27again there is one question I guess
25:40yes
25:43someone has a question
25:47I did I texted in the chat map okay is
25:51there a differential equation that is
25:53neither
25:55uh no well by definition or d e or PD it
26:01must be either PD or OD
26:03okay
26:05um well the the thing is if it's a pde
26:09it cannot be OD
26:14okay
26:18um okay all right good
26:23so there is no equation which is neither
26:25PD or nor OD it must be either an
26:29ordinary differential equation or
26:32uh partial differential equation
26:41well let's look at this one
26:45um this equation
26:47is a differential equation because we
26:49have a derivative here Y is the
26:51independent dependent variable T is the
26:54independent variable and there is only
26:57one
26:59independent variable which is T so it's
27:02an ordinary differential equation and
27:05the Order of this differential equation
27:07is one because we have only one
27:08derivative right
27:10here this is basically
27:13y Prime
27:15of T right
27:17so this gives the order
27:20and there is only one variable T is this
27:24T is
27:25uh independent
27:31variable
27:34so that means basically
27:38this is an ode
27:43of orders
27:46one
27:48okay
28:01well let's
28:03um do another example
28:08let's look at this equation
28:12the second derivative of y with respect
28:15to sometimes we call
28:17independent variable TX depends on the
28:22examples okay so we can change the
28:24independent variable and we give
28:25different letters for
28:27um
28:28for the independent variable
28:35did what you apply with respect to X
28:38uh second derivative of y Plus
28:41a d y d x
28:46to the
28:492 19
28:53plus cosine X
28:56is equal to zero
29:02well there's again one uh independent
29:05variable which is X okay and it's a
29:09second order now right
29:12so here this is the independent
29:15variables and dependent variable is y
29:17and we take the second derivative so
29:19this is order two
29:26so this is an OD
29:31of
29:33orders
29:35two see
29:43another example
29:45let's take
29:47this equation pi x x
29:52that means that we take the partial
29:54derivative with respect to X
29:58Plus
29:59cosine Cube X
30:03Phi x y
30:07square is equal to Ln
30:11x y
30:14is it an ode or PD no
30:23well now we have two variables X and Y
30:26they are independent variables and Phi
30:29is the dependent variable right we have
30:32some answer here
30:35um
30:37okay what does this sign mean I guess
30:40you're asking about this
30:45so here Phi is a function p is a
30:50function like this it depends on X and Y
30:53okay
30:54it's a function and this
30:58represents the partial derivatives okay
31:08also derivative means you keep y
31:11constant and differentiate with respect
31:14to X but don't worry about this uh we
31:18will not deal with PD partial
31:20differential equations until the end of
31:22course okay and the last two weeks or
31:25one week we will do some PD but other
31:29than that we will mostly deal with
31:32ordinary differential equations but it's
31:35good to know what is a PV so this
31:37is a PD
31:41partial differential equations because
31:43there are two
31:45independent variables okay here X and Y
31:49these are two
31:52independent
31:56variables
32:01that means it's not an OD anymore okay
32:05it's a PD what is the order
32:08well here you see we differentiate twice
32:11with respect to X
32:12and also we refresh it twice here okay
32:15so the largest derivative
32:19largest order of the derivative gives
32:21the order of the equation so it's a PDA
32:25of order two
32:30okay
32:34any question
32:41all right
32:47well let's talk about some notations
32:50these are common notations for the
32:52derivative
33:06questions
33:08well for the ant derivative derivative
33:12of order and we use this
33:18or this
33:19here Y is a function of x
33:23okay
33:24foreign
33:27oh okay there is one question why this
33:30is a PD a partial differential equations
33:32the reason is we have partial
33:34derivatives okay Phi x x means you
33:38differentiate Phi Phi here with respect
33:41to X that means we have partial
33:44derivative we keep y constant here to
33:48find that we keep y constant and we
33:50differentiate with respect to X so we
33:52have partial derivative that's why it's
33:54called partial differential equations
33:56pde okay is it clear now
34:04to be honest I wasn't able to understand
34:06because uh there are both independent
34:10variables
34:11right there is a dependent variable here
34:14as well I didn't say but this is
34:17dependent
34:20okay this one's dependent but we are
34:22taking care of its recording to X twice
34:26yes
34:27[Music]
34:28this derivative according to
34:31independence variable
34:34and
34:36[Music]
34:40um because okay let me give an example
34:42let's say Phi is
34:48well basically if you have two variables
34:51more than one variable it's called PD
34:54partial differential equation this is
34:56the definition if you have more than one
34:58independent variables then it's a
35:00partial differential equation and here
35:02we have two variables independent
35:05variables X and Y okay that's why it's
35:09PD but let me clarify this with an
35:11example let's say Phi is for instance
35:14sine
35:17x times y okay
35:21it's a function of X and Y so what is
35:23Phi sub X
35:25so what you do is to find Phi sub X you
35:29consider like Y is a constant so it
35:31becomes like cosine x times y times
35:36you consider Y is a constant so it
35:38becomes uh times y chain rule right so
35:42we differentiate with respect to X when
35:45you differentiate with respect to X that
35:48means we're talking about partial
35:49derivatives okay so it's a partial
35:52derivative
35:54and when you have a partial derivative
35:57in the equation it's called PD
36:00clear now yeah yeah thank you that makes
36:02sense all right
36:08so basically if you have more than one
36:10variable
36:11then more than one independent variable
36:14then it's called PD
36:20well don't worry about this uh we will
36:23do many examples during the course and
36:26you will get a better idea what's an OD
36:29what's an order and so on
36:33well we also have this linear 4D let's
36:37also talk about this
36:42another definition
36:47that stands for definition by the way I
36:50will use a lot in the course a
36:53differential equation
36:56at the financial equation is called
37:03a linear
37:07or the
37:11off
37:13orders
37:15and
37:17order that means
37:19we differentiate
37:21and many times right
37:24uh what what is linear means here what
37:27does linear means here well it means the
37:30following if
37:33gets off the form
37:39if it is in this form
37:41we have a function of x x is the
37:45independent variable
37:49and why is the dependent for you also
37:53does here
37:55y to the N that means and orders
37:58differential equations okay
38:02plus another function so a and X is a
38:05function it's a function of x
38:08these are called coefficients of the
38:10differential equation
38:13another function of x
38:15and we have y derivative of y of order n
38:20minus 1.
38:22I go like this
38:26A1
38:29X
38:31y Prime
38:33Plus a0
38:35X Y is equal to another function let's
38:39call it f of x so if you have a
38:42differential equation in this form it's
38:45called linear
38:47differential equation
38:49it's an ode because there is only one
38:51independent variable Let Me Maybe Call
38:54this here X is the independent variable
39:06Y is the dependent variable
39:19dependent variable
39:21okay n is the order what are this these
39:24are functions these are coefficients
39:26okay
39:27coefficients
39:34coefficient functions let's say
39:36so they are the functions of X like for
39:39instance a n minus 1 can be like x
39:41squared times cosine X
39:44any function of X okay
39:47let's give some examples
39:52is this valid in case n is not zero
39:58well if n is equal to 0 we don't have an
40:01OD
40:02maybe I should say here n is n is a
40:06natural number
40:07but if n is equal to 0
40:10we don't have a differential equation
40:13that means we don't differentiate right
40:15it's an equation it's a it's a it's a
40:18usual equation what you know from high
40:20school okay
40:30um well there is one question the
40:33coefficients different than one this is
40:35right it doesn't depend on the
40:38coefficients the only thing we require
40:40with the coefficient there must be a
40:42function of x
40:43if the coefficient doesn't depend on y
40:48then
40:50um it's a linear equation
40:52okay
40:54so this is the definition of linear uh
40:57OD
40:58the coefficients can be different from
41:01one and it's still
41:02uh linear okay
41:05let's give some examples
41:07[Music]
41:15okay
41:17okay let's look at this one
41:212x squash
41:23y double prime plus
41:26y Prime minus sine X
41:30times
41:32Y is equal to x square
41:35tangent X
41:38well is this a linear
41:42OD it's an OD for sure because there is
41:45only one independent variable
41:48but is it
41:52um
41:56is it a linear OD what do you say
42:00foreign
42:05does it look like the form here given by
42:09this we have this form here
42:12does it look like this one
42:22well there are answers no yes uh linear
42:26right thing not really a n doesn't have
42:29constant see what what is consistency
42:36okay tuba says yes
42:41well it's it's linear let's let's go
42:45back to the form here we have the
42:47coefficients they are they are the
42:49functions of X so these are
42:53functions of x
42:55functions of the independent variable
42:58okay the coefficients and see this here
43:02this is A2 for instance
43:06it's a function of x
43:09the next one is
43:11this is one times y prime one that means
43:15A1 is the constant function you can't
43:19think of this function of X again
43:22okay it's a constant function
43:26this is A1
43:34this here is a
43:36zero
43:38X okay and this is my f of x
43:43so you see here it's exactly in the in
43:46the in the form of a linear
43:49ode so it's a linear audio of order two
44:05is a linear
44:10ode of order
44:13two okay
44:23so there's one question can we assume
44:25that a2a1 a0 can be any
44:29function any coefficient yes there can
44:31be any function but they must be a
44:33function of X not a function of Y I will
44:36give an example now
44:38uh which is not maybe let's give this
44:42example
45:04y
45:06triple third derivative of y
45:10plus x times
45:21y Prime
45:25y double prime plus
45:27y times
45:30y Prime is equal to zero
45:39well is this a linear OD
45:49no it's not good we have many answers
45:52here the reason is the coefficient of Y
45:55Prime right A3 here
45:59uh it depends on y so it must be three
46:02or five so it's not a linear OD
46:07okay so because of this term
46:10okay
46:13um
46:14not
46:17a linear
46:20it's an ordinary differential equation
46:23it's an OD it's an OD of
46:27it's an OD of
46:30orders
46:31three
46:33but it's not linear
46:35okay
46:43okay there is one question can we assume
46:46that a two's order doesn't have to be no
46:49we don't care about the order of A2
46:51what's the order of A2 by the way it's a
46:55function we don't have an order for the
46:57function if it's a polynomial we can
46:59talk about its degree but it doesn't
47:01have to be polynomial the coefficients
47:04so they are functions and we don't
47:06talk about the order of a function okay
47:12cervical
47:16um
47:17[Music]
47:21right
47:23uh uh and
47:25um
47:29[Music]
47:38cannot say the coefficients at A1 a to
47:42A3 here because it's not a linear
47:44differential equation if it's a linear
47:47equation then we call the coefficients
47:48of A1 A2 A3 it's here
47:53what we say is
47:56um
47:57X
48:00is the independent variable
48:03because we take the derivative of y not
48:05X okay
48:12and Y is the independent variq
48:17right
48:21there is only one independent variable
48:23uh
48:25sorry why is it dependent for you so why
48:29is the dependent variable X is the
48:30independent variable so there's only one
48:32independent variable that's why it's OD
48:34okay
48:35okay
48:43well there's uh another question in the
48:47first example one in front of Y Prime
48:51is dependent to X because of
48:57uh yeah you can think of like this one
49:01constant function you can think of
49:02constant function of any function
49:04function of any variable okay the
49:07constant function you can't think of
49:09const margin is a function of X function
49:11of Y whatever okay it's it's free you
49:15you can you can take it as as like as
49:18you as as you want okay so you can say
49:21constant function is a function of x
49:24that's fine
49:26the thing is the coefficient cannot
49:30depend on uh here
49:33the important thing is the coefficient
49:35cannot depend on the dependent variable
49:39okay here the dependent variable is y
49:42in this equation dependent variable is y
49:45so the coefficient cannot depend on y
49:47basically
49:49or linear
49:51OD okay
49:54any other question
50:03all right
50:05um well
50:08here we have some examples of
50:10differential equations you can cook up
50:12your examples you can write many
50:14examples like this but the thing is
50:18um
50:19that doesn't mean that you can solve
50:22these equations solving equation is a
50:25totally different story okay
50:27sometimes it looks like very easy but
50:30it's incredible
50:33hard to solve differential equations so
50:35we need some certain type of
50:38differential equations to solve to deal
50:41with okay so for example of course this
50:44certain type of equations yes foreign
50:56yeah
50:59let me give another example
51:02then we will give a break
51:08um for instance y prime plus
51:112 y Square
51:14is equal to X well this is again not
51:18linear because we have y Square okay
51:29because of
51:33the term
51:37y squared
51:40it's like the coefficient of Y is like y
51:43it's you can write this like y times y
51:46so the coefficient of Y it's not a
51:49function of X it's a function of Y so
51:52um it's not linear okay
51:58right if you have a square of the
52:01derivative it's not again linear another
52:03example
52:07but say why is function of x
52:11yes
52:14so this term is
52:18we don't we don't think like that
52:22okay I understand what you're saying why
52:24is like function of X so like uh the
52:28coefficient is like function of X but
52:31I mean this is like the type of the
52:33equation if you write explicitly then
52:36but we don't know what Y is right that's
52:39the problem
52:42if you know what what is you replace y
52:44with that function and it becomes a
52:47linear
52:48audio but we don't know what it is okay
52:51so we need to see explicitly the
52:54coefficients okay
52:55in terms of maybe why it can be
52:58dependent on a second independent
53:01variable
53:04uh
53:06uh that's fine it but it's a linear PD
53:10but we don't we don't talk about this
53:12now okay it becomes a PD then
53:15now we're talking about ordinary
53:17differential equations right
53:20so for instance y double prime plus
53:25y Prime
53:27Square it's not linear again
53:35so basically you can think of this like
53:37all the derivatives
53:39appears in the differential equations
53:41without any mixed terms you don't have
53:45any squares cubes like y Cube y Prime
53:47Cube or you don't have mixed terms like
53:50y times y prime or Y primes times y
53:53double Prime so that's not
53:56let me give one more example
54:11again this is not linear because we have
54:14a mixed term the coefficient of Y Prime
54:16is
54:17it's not a function of X it's given by y
54:19okay so yes not linear
54:26they're they're all ode it's an OD of
54:30order three last example but it's not
54:32linear okay so linearities are important
54:36later we will mostly we will talk about
54:38linear audience later we will try to
54:40solve linear or these
54:42we will have several methods to solve
54:45all these linear IDs
54:48that's why it's good to know what's a
54:50linear
54:51audience okay
54:55um all right uh do you have any any
54:58other question
55:00I try to answer your questions from the
55:02chat but if I miss something you can ask
55:05now or mute yourself and ask
55:08D so as long as the coefficient of Y
55:13Primes or
55:15somewhat linked not linked X dependents
55:19even if it's like some random number
55:21like two three five whatever that's
55:24familiar that's fine yeah so so the
55:27constant functions are okay
55:30because you can't think of this right
55:33a2x is equal to five
55:37it's a you can you can it is a constant
55:40function
55:41that's fine
55:44but this is not a lot right a2x is equal
55:48to Y Square it that's that's that
55:52doesn't make sense okay that's why
55:54dependent that doesn't count
55:56right right
55:59all right let's give a break now uh for
56:0210 minutes uh so let's meet at 11 50
56:06okay
56:42um
56:55[Music]
56:56foreign
57:00[Music]
57:072016
57:17foreign
1:07:11foreign
1:07:29welcome back so before we continue do
1:07:32you have any question
1:07:34foreign
1:07:46well now let's talk about Direction
1:07:49fields
1:07:51this is 1.3 in the book
1:08:12let's take a differential equation in
1:08:14the form so take
1:08:19a differential equation
1:08:30d y d x is equal to
1:08:34f of x y
1:08:37okay
1:08:38so here X and Y are
1:08:42independent variables
1:08:45sorry X is independent variable Y is the
1:08:48dependent variable and we have here f of
1:08:51x y it's a function of X and Y
1:08:58it can be linear or not linear
1:09:02it's an OD
1:09:04okay
1:09:09well
1:09:12in this equation
1:09:15um
1:09:18the graph
1:09:23of the solution
1:09:29Y is equal to Y of x
1:09:32so Y is the solution and its graph is
1:09:35called the solution curve
1:09:43solution
1:09:45sure
1:09:49okay
1:09:52so what does it say this equation here
1:09:55we have
1:09:58they do what you will fly is equal to
1:10:01some function right hand side gives this
1:10:05derivative right derivative of 5 with
1:10:07respect to X at the point x y
1:10:11um and derivative the geometric meaning
1:10:14of the derivative is basically it's the
1:10:16tangent line the slope of the tangent
1:10:18line at that point right
1:10:21so here this f of x y
1:10:25tells you the slope of the tangent line
1:10:28to the solution curve at an arbitrary
1:10:31point x y
1:10:37so let's write this as a remark
1:10:46f x y
1:10:50it's a real number at the end right you
1:10:53give X Y so you find a real number
1:10:59gives
1:11:01the slope
1:11:09of the tangent line
1:11:13foreign
1:11:19to
1:11:21The Cure
1:11:24the solution curve Y is equal to YX
1:11:31at the point
1:11:35the curve is in R2 in the plane so we
1:11:39have two coordinates X and Y at the
1:11:41point x y
1:11:46so this is a nice fact because this will
1:11:50help to sketch the solution occur
1:11:52without finding explicit
1:11:55formula of the solution so without
1:11:59finding the solution we will able to
1:12:01draw the solution okay
1:12:04simply using this equation
1:12:07differential equation
1:12:13so how do you do that well
1:12:21so this is our goal
1:12:28the first call without finding
1:12:31without solving the equation
1:12:42the or the
1:12:45we will sketch the solution curve
1:13:01solution sure Y is equal to Y X okay
1:13:13Maybe
1:13:18this is better
1:13:20so that you can follow
1:13:38foreign
1:13:39well
1:13:45go back and let's do this
1:13:54so let's do some examples
1:14:09so let's sketch the direction field
1:14:22oh well maybe we should first talk about
1:14:25what is a directional field okay
1:14:33well let's define what's the direction
1:14:35field first then we will do some
1:14:38examples
1:14:39this well in the X Y plane to sketch the
1:14:44solution curve in the X Y plane
1:14:49uh we drove small line segments at each
1:14:53point having a slope f of x y
1:14:57um we draw
1:15:03small
1:15:06lines
1:15:09at
1:15:11the point x y
1:15:15with
1:15:16slope
1:15:19equal to
1:15:21f of x y
1:15:26and the curve following those lines
1:15:30are basically the solution curve and the
1:15:33sketches called the direction field okay
1:15:37um
1:15:39this sketch is called Direction field
1:15:57reaction field
1:15:59and if you're following this small line
1:16:02segments it's called the it's the
1:16:05solution cure
1:16:09with curves
1:16:12solution is not unique so we have many
1:16:15solution curves we'll talk about this as
1:16:17well
1:16:18slowly the curves following
1:16:25these lines
1:16:33are the solution curves
1:16:40okay
1:16:52okay let's do one example with this
1:16:59an easy one uh let's
1:17:03take this equation
1:17:08d y d x is equal to 2x this is quite
1:17:11easy to solve but we will not solve it
1:17:14now later we will solve it
1:17:16[Music]
1:17:17um
1:17:20sketch the direction field
1:17:35and
1:17:37solution curves
1:17:52all right well
1:17:55we work in the plane so let's
1:17:59draw
1:18:02a plane like this
1:18:07to draw the direction field we take some
1:18:11arbitrary points uh
1:18:14there is no rule how to choose these
1:18:17points you can choose
1:18:19as much as you as many as you want okay
1:18:23so let's take some points like this
1:18:30these are like lattice points so we have
1:18:33one two three for instance
1:18:41okay
1:19:03so these are some points with integer
1:19:06coefficients integer coordinates
1:19:10minus one minus two minus three and so
1:19:15on this is zero
1:19:18all right so let's uh try to sketch the
1:19:21direction field well at each point for
1:19:25instance we are at this point let's say
1:19:27at one the x coordinate is one and our
1:19:31function is this is your f of x y
1:19:36so the slope at this point is 2 okay so
1:19:402 is like this
1:19:43[Music]
1:19:44um
1:19:45so you will draw a
1:19:47line with slop two it's like
1:19:51this one okay
1:19:58and maybe let's go to that point
1:20:02oh wait wait this is one this is with
1:20:04slog one
1:20:07and
1:20:08let's go to this point this point is is
1:20:12like we have like slope this is two two
1:20:16two right it's two two
1:20:19so the function here f of x y it takes
1:20:24um right this is slope two is it's going
1:20:28to be slope four it's gonna be
1:20:30more like this it's like more vertical
1:20:33right
1:20:39now this is X and Y
1:20:48and when you move right hand side it's
1:20:52gonna be like more and more vertical so
1:20:56for instance at three it's like this
1:21:04okay
1:21:06at two it's like this
1:21:11um well I know this is not a very good
1:21:13picture
1:21:24they're all alike like this
1:21:27with the same x coordinate
1:21:29okay
1:21:32at one maybe let's
1:21:39do it like this
1:21:44okay
1:21:48so at zero it is zero
1:21:51okay
1:22:02well we cannot okay there is a question
1:22:04how accurate we should draw this there
1:22:07is uh you don't have to be
1:22:10too much accurate with this okay so
1:22:13just follow the you know basic uh
1:22:17geometry and it doesn't have to be
1:22:19perfect picture I would say
1:22:28so why do we choose two two we don't
1:22:31choose two two here we we this is just
1:22:33an example okay which we we drove this
1:22:37line segments at each point okay each
1:22:40points like this
1:22:42okay tutu was just an example
1:22:46okay maybe let's start this again
1:22:57for instance at
1:23:04one zero
1:23:07the slope is what
1:23:11F of 1 0 which is two okay
1:23:17so it looks like this
1:23:20two is like
1:23:22um
1:23:22some like this
1:23:25okay
1:23:27at
1:23:292 0 for instance
1:23:36the slope is 4.
1:23:42f 2 0 which is
1:23:46four
1:23:50so it's going to be like
1:23:53more vertical
1:23:54like this
1:23:56okay
1:23:58at 2 0 at 2 1
1:24:02at 2 2
1:24:04they are same
1:24:05right they have the same slope because
1:24:08it doesn't depend on the y coordinate so
1:24:10we have the same slope like this
1:24:13like this
1:24:16like this okay
1:24:20yeah
1:24:22at zero to zero
1:24:24that means we have
1:24:28a horizontal
1:24:31like this
1:24:33the slope is zero
1:24:36so let's continue at one
1:24:39we had the same slope with 1 1 so it's
1:24:42going to be their parallel okay like
1:24:44this
1:24:49okay
1:24:55well
1:24:56at three it's going to be more vertical
1:24:58it's going to be like
1:25:00more like this
1:25:08okay
1:25:11and the same what happens on the left
1:25:12hand side
1:25:14minus one it's going to be
1:25:18um let's take a point for instance
1:25:21minus one
1:25:24zero
1:25:27we have the slope
1:25:30minus two
1:25:32okay
1:25:34minus two means something like that
1:25:43okay
1:25:45at minus two zero the slope is
1:25:50this
1:25:51minus Forge
1:25:53so it's going to be more vertical
1:25:57like this
1:26:03okay so let's
1:26:05um okay what about on the y-axis well
1:26:10it's always like horizontal
1:26:14because x is 0 here
1:26:18so we have sort of a picture right so
1:26:21this is the direction field let's draw
1:26:23the solution curves
1:26:26um
1:26:27so what we have is like okay
1:26:33let's follow for instance
1:26:36uh dispatch
1:26:47okay
1:26:50so you see these lines are like tangent
1:26:52to this curve
1:26:55another curve is like this
1:27:02uh it's not a good drawing but it must
1:27:06be like maybe more like this
1:27:11okay
1:27:12and um
1:27:16you can draw another one like like this
1:27:22Okay so
1:27:26so these are solution cures okay
1:27:37uh any question with this
1:27:43yeah if so when you go to Infinity with
1:27:47x
1:27:49right so the the slope will be Infinity
1:27:51that means it's going to be a vertical
1:27:53line
1:27:57and that makes sense because this now we
1:28:00will solve this and you will see why
1:28:02this happens okay
1:28:04so if you solve this equation how do you
1:28:07solve this equation by the way this
1:28:09differential equations
1:28:12how do you solve this
1:28:24so there's one question why do we have
1:28:27this horizontal lines at the origin at
1:28:29the origin X is equal to zero
1:28:33okay
1:28:34on the y-axis X is equal to zero and f
1:28:38of x y is equal to 2x 2 times 0 is 0.
1:28:41that means the slope is zero so we have
1:28:43a horizontal tangent line right
1:28:46well right so we integrate we integrate
1:28:50this
1:28:51so we had
1:28:57d y idx is equal to 2X
1:29:02and integrate both sides
1:29:09like this
1:29:24or maybe let's run like this
1:29:29it's a better
1:29:32oops
1:29:45okay and now we get what we get
1:29:48basically
1:29:50Y is equal to x square over 2 plus a
1:29:55constant
1:30:02so you see here x square over 2 is a
1:30:04parabola which looks like this here
1:30:06solution curves right
1:30:09and as X goes to Infinity right so it's
1:30:12going to be like
1:30:13the tangent line will be more and more
1:30:15vertical so it's going to be a vertical
1:30:17line at the infinity
1:30:20the the C here is arbitrary C is any
1:30:24constant
1:30:26and you can also see here in the
1:30:29solution curves when C is equal to zero
1:30:31this is the solution curve the first one
1:30:35so this is one
1:30:38C is equal to zero
1:30:41here this is when C is equal to 1.
1:30:45okay so you have many solutions you have
1:30:48infinitely many solutions
1:30:50depending on the constant in the
1:30:52solution
1:30:53it was an easy differential equation so
1:30:55we could able to solve it this is not
1:30:58the case always
1:30:59okay
1:31:01but still we can draw the direction
1:31:04field
1:31:06using the equation without solving okay
1:31:09any question with this
1:31:19all right this is
1:31:24gonna be x square sorry
1:31:31all right thanks
1:31:33well let's do another example
1:31:44here we consider
1:31:47um
1:31:48this one y Prime is equal to Y times
1:31:53y minus 1.
1:31:56okay
1:32:02well first of all this is not a linear
1:32:04body but
1:32:06that's not the point here
1:32:12um
1:32:13well let's say this is my
1:32:21F of x y
1:32:25so that means basically the slope is
1:32:28given again
1:32:29right hand side because this is the lip
1:32:31on the left hand side we have to deal
1:32:33with two and derivative gives the slope
1:32:34of the tangent why
1:32:36so
1:32:38let's draw the sketch Direction field
1:32:41first
1:32:43or maybe well this is not easy now
1:32:46what I will do is I will do in a more
1:32:49systematic way here
1:32:51um
1:32:53so let's try to understand what happens
1:32:56with the slope okay the slope is given
1:32:58by a function y times y minus 1. so
1:33:01let's make a table for this purpose but
1:33:04uh
1:33:06we also want to know the concavity of
1:33:09the solution curve so for this purpose
1:33:11let's look at the second derivative
1:33:13remembering calculus
1:33:15to get some information about the
1:33:17concavity we look at the second
1:33:18silhouette right concave concave down if
1:33:22it is the second derivative is positive
1:33:24it is going to give you up if it is
1:33:26negative it is concave down and so on
1:33:28right
1:33:32for concavity
1:33:43we check
1:33:49second derivative
1:33:52well what we do is we differentiate this
1:33:55again
1:33:56from here we get what we get basically
1:34:00the second derivative of y
1:34:04differentiate with respect to X
1:34:09we use the product rule y Prime Times y
1:34:13minus 1
1:34:17plus y times
1:34:20y minus
1:34:22y times y Prime so
1:34:37well
1:34:41um let's write this everything in y so
1:34:44it will place y Prime by
1:34:47y times y minus 1 so it becomes
1:34:55uh y times
1:35:00y minus 1. Square
1:35:04because y Prime is equal to Y times y
1:35:07minus 1 plus
1:35:09y times
1:35:11y times y minus 1 y Square Times y minus
1:35:161.
1:35:18or you can take those in parentheses and
1:35:21it becomes
1:35:22um
1:35:24y times y minus 1 parentheses we have
1:35:28y minus 1
1:35:31plus y two y minus 1.
1:35:35this
1:35:37so for concavity we will use this
1:35:52okay there's one question how do you
1:35:54obtain this well
1:35:56yeah
1:35:59you see y Prime is given by this so we
1:36:01replace y Prime with this okay
1:36:05we replace y Prime by right hand side
1:36:18all right let's make a table now
1:36:31we have y
1:36:34or maybe that's okay when Y is equal to
1:36:370
1:36:391 and 1 half we have the roots
1:36:43and also just for the first derivative
1:36:46or the same 0 and 1.
1:36:48one Health and one well we have
1:36:56the first derivative
1:36:58we have the second derivative
1:37:01and we also have y
1:37:11foreign
1:37:18[Music]
1:37:25y times y minus one so
1:37:29um
1:37:31it's positive negative negative positive
1:37:35right
1:37:39second derivative uh we have
1:37:44all the roots are simple Roots so we
1:37:47have positive
1:37:49they're all zeros right
1:37:52negative positive negative
1:37:55for the second derivative so that means
1:37:57what the first derivative positive means
1:37:59it's an increasing function y is an
1:38:02increasing function
1:38:04second derivative is negative means it's
1:38:06a concave down right so
1:38:09increasing and concave down it looks
1:38:11like this
1:38:15increasing plus concave down right here
1:38:22so from zero to one Health it's
1:38:25decreasing because derivative is
1:38:27negative and concave up so decreasing
1:38:31and concave
1:38:33up
1:38:34it's like that
1:38:39from one half to one it's
1:38:43still uh decreasing
1:38:45but now it's concave down so it's like
1:38:49this
1:38:51when Y is bigger than one
1:38:54it's increasing and concave up so it's
1:38:57like this
1:39:00so the solution Curve will look like
1:39:02this basically but let's sketch this now
1:39:06put all this information
1:39:19okay let's
1:39:44okay so the the critical points are like
1:39:47when Y is equal to here
1:39:51zero okay uh when Y is equal to one
1:39:55Health
1:39:56which is here
1:40:05and when Y is equal to 1 we have this so
1:40:10this is X and Y
1:40:17okay
1:40:19so what happens here is from
1:40:22when Y is uh bigger than one
1:40:28maybe I should take this down a little
1:40:31bit
1:40:43foreign
1:40:52okay when Y is greater than one
1:40:56um
1:40:58we are here right this one it's
1:41:01increasing
1:41:03and concave up so the Curve will look
1:41:06like this
1:41:14well
1:41:16you may have of course many curves like
1:41:20this right because solution is not
1:41:22unique like in the previous example so
1:41:25basically we draw several curves like
1:41:28this in the same uh
1:41:31manner but
1:41:33depends it depends on you how many you
1:41:36draw let's draw three of them for
1:41:38instance we have
1:41:40another one depending on the constants
1:41:44right
1:41:45constants in the solution so we have
1:41:48three curves like this let's say you can
1:41:50do you can draw three two curves depends
1:41:54on you
1:41:56when why in one Health and one
1:41:59well it's a decreasing function and
1:42:02concave
1:42:04down so it looks like basically
1:42:09um
1:42:12like this
1:42:15from zero to one Health it's still
1:42:19decreasing but it becomes concave up so
1:42:21it's like this
1:42:25like this okay
1:42:28and again we draw several lines like
1:42:30this
1:42:41all right
1:42:45when Y is less than zero
1:42:48what happens with the solution curve
1:42:50well
1:42:52uh when Y is less than zero it is
1:42:54increasing and concave down so it's
1:42:57going to be like
1:42:59just increasing and concave down
1:43:02they will look like this curves okay
1:43:07so these are the solution curves
1:43:09basically
1:43:16any question with this
1:43:23um so
1:43:24couldn't could have we gotten the same
1:43:26result using the direction field instead
1:43:30yeah you could do it but it's hard to
1:43:34guess
1:43:35the precise uh picture the solution
1:43:39curves but if you draw uh the precise uh
1:43:43slopes you will get the same picture for
1:43:46sure
1:43:47but that's not easier that's why we
1:43:50follow this way
1:43:51another thing
1:43:53um say we have
1:43:56a very similar picture but some higher
1:43:59or some higher order
1:44:01say like instead of Y Prime we got y
1:44:05to the prime four for example that
1:44:09in that case uh you I mean depending on
1:44:12the equation you can still draw with but
1:44:15uh you need to know the first derivative
1:44:18second derivative if you can derive
1:44:20first derivative and second derivative
1:44:22from the equation
1:44:24then you can draw it right because this
1:44:27is what we did here uh we know the first
1:44:30derivative using first derivative we
1:44:32find a second derivative
1:44:34uh but if you know only second
1:44:37derivative uh you don't know the first
1:44:40derivative and you cannot draw the the
1:44:43similar picture Okay so as long as we
1:44:46can you know extrapolate the first and
1:44:48second derivatives from the picture
1:44:49right right that's right that's right if
1:44:52you can extract second derivative and
1:44:53first derivative from the equation then
1:44:55you can't
1:44:59thanks good
1:45:10okay there's one question uh why they
1:45:14are studying
1:45:16minus infinity and go to plus infinity
1:45:20uh well
1:45:22this follows from the table so for
1:45:25instance when Y is look at this part
1:45:32would get this part okay this is when Y
1:45:35is bigger than one and when Y is bigger
1:45:38than one we have this sort of behavior
1:45:41so y must be bigger than one so you must
1:45:44be above the line
1:45:46the horizontal line here you must be
1:45:49about this
1:45:51and it must increase increase means from
1:45:55left to right you go up okay because of
1:45:58this picture here
1:46:02that's why it's it goes from like minus
1:46:05infinity to plus infinity goes up okay
1:46:08so it follows from the table
1:46:15well we draw this horse okay there is
1:46:18one question why do we need those
1:46:20horizontal lines without this horizontal
1:46:23lines it's hard to draw the solution
1:46:25curve this will help to draw the uh
1:46:28solution curve but it's not a part of
1:46:31the solution
1:46:32final solution I would say you can
1:46:35delete for instance
1:46:38the horizontal lines at the end but it
1:46:41they will help
1:46:42just to sketch the solution curves okay
1:46:46they are not necessary but without them
1:46:49it's not easy to do
1:46:54okay so let's maybe we have one one
1:46:57minute only so let's
1:47:00go back to the table there's one
1:47:02question about the table how we get this
1:47:04table well this table here we have y y
1:47:08Prime y second prime
1:47:10the first derivative gives this the
1:47:13information about increasing decreasing
1:47:16if the first derivative is positive it's
1:47:18increasing otherwise it is decreasing
1:47:20right
1:47:21so the the roots of the first derivative
1:47:24we use
1:47:26this equation the first derivative first
1:47:29derivative is 0 when Y is zero and one y
1:47:32is one
1:47:33so these are the zeros of the first
1:47:36derivative they are all single Roots
1:47:39they don't repeat okay so that's why we
1:47:42have plus minus minus plus okay
1:47:45between two Roots the sign will change
1:47:49the second derivative the roots of the
1:47:51second derivative are here zero one and
1:47:54one Health so they're all the zeros of
1:47:57here this one
1:48:00there's one on that one they are the
1:48:02zeros of the second derivative
1:48:04so we have plus minus at each root we
1:48:08change the sign
1:48:10when the second derivative is
1:48:12negativities concave down if when it is
1:48:16positivities concave up
1:48:18so depending on this we have this
1:48:20picture
1:48:21for instance this was decreasing and
1:48:25concave up
1:48:27okay
1:48:29and using this table we draw this
1:48:32solution curves all right
1:48:35any other question or comment
1:48:39okay we will stop here
1:48:42uh well there's one one more question
1:48:44why do we draw three three is not
1:48:47important you can draw one two three
1:48:49four five as many as you want
1:48:52uh
1:48:54why don't we draw just one because the
1:48:57solution curve is not a unique curve
1:48:59remember in the previous example we had
1:49:02a constant C right x squared plus C that
1:49:06means depending on C we have infinitely
1:49:09many curves and we draw several of them
1:49:11to just to emphasize that the solution
1:49:14is not a unique curve okay
1:49:19all right I'll stop here
1:49:22uh I will put this recording and lecture
1:49:25not so not the class today
1:49:28um see you next lecture on Friday
1:49:31morning bye
1:49:36Fighters
1:49:38bye

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