1 00:00:00,571 –> 00:00:06,006 Hello and welcome back to the video series about ODEs

2 00:00:06,314 –> 00:00:10,578 and in today’s part 3 we will talk about the directional field.

3 00:00:10,778 –> 00:00:16,886 This one will help us here at the beginning of this topic to visualize ODEs.

4 00:00:17,257 –> 00:00:23,204 Indeed it’s a vector field that can be sketched such that we understand the problem.

5 00:00:23,404 –> 00:00:26,228 Ok, but before we start with that, you already know,

6 00:00:26,428 –> 00:00:31,982 first I want to thank all the nice people who support this channel on Steady via Paypal or by other means.

7 00:00:32,414 –> 00:00:39,625 In fact you should know only because of your support it’s possible to create such videos about mathematics

8 00:00:39,825 –> 00:00:45,723 and as a thank you and as a bonus you can download the PDF version and the quiz for this video.

9 00:00:46,300 –> 00:00:51,066 Ok, then let’s start by telling again what we mean by an ODE.

10 00:00:51,266 –> 00:01:00,245 So you might remember if we write x dot (the first derivative of x) is equal to a function w, then we have an ODE.

11 00:01:00,614 –> 00:01:06,888 Moreover this function w has the independent variable t as an input and x itself.

12 00:01:07,357 –> 00:01:15,265 In addition you should know that x can be a vector. So we can have a whole system of ODEs.

13 00:01:15,465 –> 00:01:24,051 However to be more precise you have already learned that an ODE of this form is called explicit and of first order.

14 00:01:24,600 –> 00:01:32,428 This means that we only have the first derivative in the whole equation and the first derivative is also alone on the left-hand side.

15 00:01:32,743 –> 00:01:39,747 However, I can already tell you that in the next video we will show that this is not really a restriction.

16 00:01:40,243 –> 00:01:45,924 Indeed, we will see that we can simplify that even more without losing generality

17 00:01:46,124 –> 00:01:50,268 and in order to see that let’s first look at some examples here

18 00:01:50,829 –> 00:01:56,437 and the first one here should be the ODE x dot is equal to lambda times x.

19 00:01:56,743 –> 00:02:03,945 So a very common example, but what we should see here is that the function w doesn’t depend on t.

20 00:02:04,145 –> 00:02:09,769 Hence the independent variable, the time variable, does not occur in the equation.

21 00:02:09,969 –> 00:02:15,787 Therefore we introduce a new name for such ODEs. They are called autonomous.

22 00:02:16,286 –> 00:02:21,619 On the contrary we can immediately think of an example which is not autonomous.

23 00:02:22,229 –> 00:02:25,415 Namely x dot is equal to t.

24 00:02:25,700 –> 00:02:30,561 There the first variable t will change the value of the function w.

25 00:02:31,200 –> 00:02:35,983 Hence this one is an ODE, but not an autonomous ODE

26 00:02:36,329 –> 00:02:40,871 and of course this notion can also be used for a system of ODEs.

27 00:02:40,986 –> 00:02:43,525 For example in 2-dimensions.

28 00:02:44,100 –> 00:02:48,708 So here x dot is equal to (x_2, -x_1).

29 00:02:49,300 –> 00:02:56,415 So you see, there is no t on the right-hand side here. Hence the whole system is also called autonomous.

30 00:02:56,971 –> 00:03:03,744 Indeed, exactly these systems will be the interesting ones so let’s consider them for the rest of the video

31 00:03:04,214 –> 00:03:08,005 and of course we have to put this into a new definition.

32 00:03:08,386 –> 00:03:12,942 In fact we want to keep it short and just call it an autonomous system.

33 00:03:13,614 –> 00:03:20,457 So it means we have a system of ODEs and the function w on the right-hand side does not depend on t.

34 00:03:20,786 –> 00:03:25,644 Therefore we can just write it as a function w only depending on x.

35 00:03:26,129 –> 00:03:34,945 Moreover you know we have a whole system, so this can be a vector equation. Which means the function v maps R^n to R^n.

36 00:03:35,471 –> 00:03:41,024 However in general you know v can have a domain so it maps U into R^n,

37 00:03:41,486 –> 00:03:45,864 but of course U should then be a subset of R^n as well

38 00:03:46,357 –> 00:03:50,556 and indeed often we will choose U as an open set.

39 00:03:50,756 –> 00:03:58,716 Moreover I would also say that all the times you see that v is at least a continuous function defined on U.

40 00:03:59,100 –> 00:04:04,640 Ok, but now the point is, such an autonomous system can be nicely visualized

41 00:04:04,840 –> 00:04:08,629 and this is done by the so called directional field.

42 00:04:09,086 –> 00:04:15,331 I would say this is easy to understand, because it’s simply a plot of our function v.

43 00:04:15,614 –> 00:04:20,870 So you could say this is a vector field which visualizes the function v.

44 00:04:21,070 –> 00:04:26,232 Therefore you would start with the domain U. Which is a subset of R^n.

45 00:04:26,432 –> 00:04:30,333 So for example it could be the whole R^2.

46 00:04:30,400 –> 00:04:36,267 Hence here let’s visualize an example where v maps R^2 into R^2.

47 00:04:36,886 –> 00:04:44,937 So if you choose any point p in this coordinate system, the resulting value of v will also be a vector

48 00:04:45,300 –> 00:04:50,911 and this vector we can simply visualize as an arrow starting at p.

49 00:04:51,111 –> 00:04:55,552 Now, this is a common way to explain a vector function like v

50 00:04:55,752 –> 00:05:02,331 and now of course the important fact here is that you could do it at any other point in this coordinate system

51 00:05:03,114 –> 00:05:07,052 and then you simply have a picture with a lot of arrows.

52 00:05:07,800 –> 00:05:14,761 However we now call it the directional field of the ODE, because it explains what solutions have to do.

53 00:05:14,961 –> 00:05:24,790 Namely the ODE, the whole system of ODEs says that at a given point this vector is exactly the derivative of the solution.

54 00:05:25,400 –> 00:05:31,179 Indeed this fits nicely in with the pictures of the orbits we had in the last video.

55 00:05:31,379 –> 00:05:36,961 Please note, an orbit would be the image of a solution here in our domain.

56 00:05:37,557 –> 00:05:47,257 However, now we know this can only be a solution of our system if at each point the vector we see is a tangent for this curve.

57 00:05:47,900 –> 00:05:54,849 Simply because at each point the vectors here tell us in which direction the solution should flow.

58 00:05:55,049 –> 00:05:59,671 In other words if we plot the directional field of this vector function v,

59 00:05:59,757 –> 00:06:04,171 we can already see what the solutions can do and what they can’t do.

60 00:06:04,714 –> 00:06:12,508 Therefore this whole thing here is already very helpful to see the behaviour of possible solutions of the ODE,

61 00:06:12,708 –> 00:06:17,075 but of course this whole procedure here, we should explain with examples

62 00:06:17,657 –> 00:06:21,354 and first let’s start with a 1-dimensional example.

63 00:06:21,554 –> 00:06:25,163 So x dot should be equal to the sine of x.

64 00:06:25,457 –> 00:06:31,305 So not too complicated, but we also realize that we don’t see all the solutions immediately.

65 00:06:31,714 –> 00:06:37,521 However we can recognize that v is defined on the whole real number line

66 00:06:37,721 –> 00:06:42,904 and this means that our directional field is now just a 1-dimensional picture.

67 00:06:43,471 –> 00:06:48,944 Hence for the arrows we want to draw in there are only 2 possible directions.

68 00:06:49,614 –> 00:06:56,343 To see what happens here let’s look at some important points like 0, pi/2, pi and so on.

69 00:06:56,543 –> 00:07:01,709 Of course we choose exactly these numbers, because of the sine function.

70 00:07:01,909 –> 00:07:08,542 So for example if we put pi/2 into our function v, we get out 1.

71 00:07:08,742 –> 00:07:13,757 In other words we have an arrow of length 1 that points to the right.

72 00:07:14,329 –> 00:07:24,341 On the other hand if we choose 3/2 times pi and put that into our function v, we get out -1. So an arrow that points to the left.

73 00:07:24,929 –> 00:07:31,508 So you see this is what we could do for all points on the real number line and then we get the whole directional field.

74 00:07:32,171 –> 00:07:40,891 So as another example if you choose a small, positive number here on the right of 0, we also get an arrow that points to the right.

75 00:07:41,543 –> 00:07:48,081 However please note at the point 0 and the point pi, we simply get out 0.

76 00:07:48,281 –> 00:07:53,046 This means at these points, there is no arrow pointing anywhere,

77 00:07:53,486 –> 00:07:59,029 but exactly that is immediately an important information for the solutions.

78 00:07:59,586 –> 00:08:05,100 So for example you could take a solution alpha(t) that is always 0.

79 00:08:05,600 –> 00:08:09,996 So it means alpha(t) vanishes no matter which t you put in

80 00:08:10,196 –> 00:08:14,151 and now we can simply check that this is indeed a solution.

81 00:08:14,351 –> 00:08:19,228 This is not hard to do. You simply put alpha into the ODE.

82 00:08:19,428 –> 00:08:24,718 So we want that alpha-dot(t) is equal to sin(alpha(t)).

83 00:08:25,857 –> 00:08:35,824 So this is what a solution has to fulfill for all t and here we don’t have a problem, because the derivative is 0 and sin(0) is also 0.

84 00:08:36,571 –> 00:08:40,596 So here we simply have 0 = 0.

85 00:08:41,100 –> 00:08:48,927 Now, this is important to remember. This always works if you find a point in the directional field that has no arrow at all

86 00:08:49,400 –> 00:08:54,006 and therefore we can immediately give a second solution alpha(t).

87 00:08:54,486 –> 00:08:58,651 We simply choose it as the constant function will value pi.

88 00:08:58,851 –> 00:09:06,799 Indeed, with the same reasoning this is also a solution, because the derivative is 0 and sin(pi) is 0

89 00:09:07,314 –> 00:09:11,591 and the important fact is that this claim holds for all t in R.

90 00:09:12,429 –> 00:09:19,130 In other words you see here that we have solutions that stay with their orbit just on one point

91 00:09:19,657 –> 00:09:24,467 and for this reason these points in the picture are called stationary points.

92 00:09:25,200 –> 00:09:30,406 However the directional field does not only tell us about the constant solutions here.

93 00:09:31,214 –> 00:09:37,499 Moreover you should see that we also get information about the solution that lies in here.

94 00:09:37,699 –> 00:09:42,717 In particular we can look at a solution that hits the value pi/2.

95 00:09:42,917 –> 00:09:51,686 There we know if we now increase t, we have to go to the right with the solution, because this is the direction of the arrow.

96 00:09:52,057 –> 00:09:57,362 Indeed, this is the direction of all the arrows in the interval 0 to pi.

97 00:09:58,057 –> 00:10:05,816 Therefore we can directly conclude that a solution inside this interval has to be monotonically increasing.

98 00:10:06,186 –> 00:10:14,054 Moreover we also see that we can’t exceed the value pi in the end, because pi is a stationary point.

99 00:10:14,829 –> 00:10:20,244 This means if we go with t to infinity, we will approach this value pi.

100 00:10:20,514 –> 00:10:27,326 So this is very nice. Without knowing the explicit solution, we can already say something about its behaviour.

101 00:10:28,043 –> 00:10:30,920 So this is the big advantage of the directional field.

102 00:10:31,120 –> 00:10:34,483 It helps you immediately with understanding solutions.

103 00:10:34,683 –> 00:10:40,736 However maybe the whole thing is more interesting if we look at a higher dimensional example.

104 00:10:41,586 –> 00:10:46,883 So I would say let’s take a 2-dimensional one and a similar one to the last video.

105 00:10:47,486 –> 00:10:53,306 So x_1 dot should be -x_2 and x_2 dot should be x_1.

106 00:10:53,829 –> 00:10:58,395 So here our function v goes from R^2 into R^2.

107 00:10:58,929 –> 00:11:04,103 Hence there the directional field can be visualized in a 2-dimensional plane.

108 00:11:04,757 –> 00:11:10,634 Each point in the plane get’s mapped to a vector which is given by (-x_2, x_1).

109 00:11:11,186 –> 00:11:16,247 Therefore let’s draw the plane and let’s visualize this directional field.

110 00:11:16,814 –> 00:11:24,203 Indeed, we see that this whole picture is not so complicated, because essentially it’s just flipping the arrows.

111 00:11:24,957 –> 00:11:30,686 Therefore the arrows also get longer if we get further away from the origin.

112 00:11:31,300 –> 00:11:37,478 Of course now we could fill in the whole picture and what you should see is such a rotational field.

113 00:11:38,014 –> 00:11:42,129 Now, if you don’t want to draw it by hand, you can also use a computer.

114 00:11:42,178 –> 00:11:45,753 For example Python will make this nice picture.

115 00:11:46,529 –> 00:11:52,354 There please note, I have shortened the arrows such that you can see this nice flow here.

116 00:11:52,886 –> 00:12:00,121 In other words, if we have a solution, it will go around the origin. Namely in the positive direction.

117 00:12:00,643 –> 00:12:05,641 That simply because we now know, a solution has to follow the arrows.

118 00:12:06,243 –> 00:12:11,285 Hence in our example here, the orbits will look like circles.

119 00:12:12,029 –> 00:12:18,899 So the result is the same as before. We can already sketch solutions with the help of the directional fields.

120 00:12:19,357 –> 00:12:25,665 Moreover we also see that we have a stationary solution, a stationary point in the picture.

121 00:12:26,286 –> 00:12:29,902 Indeed, there is only one and it is the origin.

122 00:12:30,729 –> 00:12:36,967 Ok, so in summary you see, the directional field helps us a lot to visualize solutions.

123 00:12:37,486 –> 00:12:44,320 However some questions remain. What does it tell us about the existence and uniqueness of solutions?

124 00:12:45,071 –> 00:12:49,447 And I tell you, these questions we will answer with the next videos.

125 00:12:49,971 –> 00:12:54,200 So I really hope we meet again and have a nice day. Bye!

Q1: Is the ODE given by $$ \cos(\dot{x}) + (\ddot{x})^4 + \dddot{x} + t^5 = 5$$ autonomous?

A1: Yes!

A2: No!

A3: One needs more information.

Q2: Is the equation given by $$ \dot{x} = x^3 $$ an autonomous ordinary differential equation?

A1: Yes!

A2: No!

A3: One needs more information.

Q3: What is the directional field of the ODE $$ \dot{x} - \sin(x) = x$$ as a function $v: \mathbb{R} \rightarrow \mathbb{R}$?

A1: $v(x) = \sin(x) + x$

A2: $v(x) = -\sin(x) + x$

A3: $v(x) = \sin(x) - x$

A4: $v(x) = x$