1
00:00:00,000 --> 00:00:00,640


2
00:00:00,640 --> 00:00:03,180
I think it's reasonable to do one more separable

3
00:00:03,180 --> 00:00:07,214
differential equations problem, so let's do it.

4
00:00:07,214 --> 00:00:13,429
The derivative of y with respect to x is equal to y

5
00:00:13,429 --> 00:00:22,460
cosine of x divided by 1 plus 2y squared, and they give us

6
00:00:22,460 --> 00:00:27,080
an initial condition that y of 0 is equal to 1.

7
00:00:27,079 --> 00:00:29,919
Or when x is equal to 0, y is equal to 1.

8
00:00:29,920 --> 00:00:32,130
And I know we did a couple already, but another way to

9
00:00:32,130 --> 00:00:34,530
think about separable differential equations is

10
00:00:34,530 --> 00:00:37,300
really, all you're doing is implicit

11
00:00:37,299 --> 00:00:39,000
differentiation in reverse.

12
00:00:39,000 --> 00:00:41,929
Or another way to think about it is whenever you took an

13
00:00:41,929 --> 00:00:46,340
implicit derivative, the end product was a separable

14
00:00:46,340 --> 00:00:47,800
differential equation.

15
00:00:47,799 --> 00:00:52,129
And so, this hopefully forms a little bit of a connection.

16
00:00:52,130 --> 00:00:53,030
Anyway, let's just do this.

17
00:00:53,030 --> 00:00:55,480
We have to separate the y's from the x's.

18
00:00:55,479 --> 00:00:59,169
Let's multiply both sides times 1 plus 2y squared.

19
00:00:59,170 --> 00:01:07,049
We get 1 plus 2y squared times dy dx is equal

20
00:01:07,049 --> 00:01:10,359
to y cosine of x.

21
00:01:10,359 --> 00:01:13,129
We still haven't fully separated the y's and the x's.

22
00:01:13,129 --> 00:01:17,189
Let's divide both sides of this by y, and then let's see.

23
00:01:17,189 --> 00:01:23,079
We get 1 over y plus 2y squared divided by y, that's

24
00:01:23,079 --> 00:01:31,019
just 2y, times dy dx is equal to cosine of x.

25
00:01:31,019 --> 00:01:34,429
I can just multiply both sides by dx.

26
00:01:34,430 --> 00:01:42,860
1 over y plus 2y times dy is equal to cosine of x dx.

27
00:01:42,859 --> 00:01:45,379
And now we can integrate both sides.

28
00:01:45,379 --> 00:01:50,670


29
00:01:50,670 --> 00:01:54,090
So what's the integral of 1 over y with respect to y?

30
00:01:54,090 --> 00:01:57,620
I know your gut reaction is the natural log of y, which is

31
00:01:57,620 --> 00:02:01,060
correct, but there's actually a slightly broader function

32
00:02:01,060 --> 00:02:03,450
than that, whose derivative is actually 1 over y, and that's

33
00:02:03,450 --> 00:02:06,549
the natural log of the absolute value of y.

34
00:02:06,549 --> 00:02:11,639
And this is just a slightly broader function, because it's

35
00:02:11,639 --> 00:02:15,549
domain includes positive and negative numbers, it just

36
00:02:15,550 --> 00:02:16,240
excludes 0.

37
00:02:16,240 --> 00:02:18,909
While natural log of y only includes

38
00:02:18,909 --> 00:02:21,159
numbers larger than 0.

39
00:02:21,159 --> 00:02:23,750
So natural log of absolute value of y is nice, and it's

40
00:02:23,750 --> 00:02:27,199
actually true that at all points other than 0, its

41
00:02:27,199 --> 00:02:28,989
derivative is 1 over y.

42
00:02:28,990 --> 00:02:31,140
It's just a slightly broader function.

43
00:02:31,139 --> 00:02:33,549
So that's the antiderivative of 1 over y, and we proved

44
00:02:33,550 --> 00:02:35,330
that, or at least we proved that the derivative of natural

45
00:02:35,330 --> 00:02:38,000
log of y is 1 over y.

46
00:02:38,000 --> 00:02:40,889
Plus, what's the antiderivative of 2y with

47
00:02:40,889 --> 00:02:41,409
respect to y?

48
00:02:41,409 --> 00:02:45,020
Well, it's y squared, is equal to-- I'll do the

49
00:02:45,020 --> 00:02:47,230
plus c on this side.

50
00:02:47,229 --> 00:02:48,989
Whose derivative is cosine of x?

51
00:02:48,990 --> 00:02:50,240
Well, it's sine of x.

52
00:02:50,240 --> 00:02:53,290


53
00:02:53,289 --> 00:02:56,650
And then we could add the plus c.

54
00:02:56,650 --> 00:02:59,340
We could add that plus c there.

55
00:02:59,340 --> 00:03:00,870
And what was our initial condition? y of

56
00:03:00,870 --> 00:03:02,000
0 is equal to 1.

57
00:03:02,000 --> 00:03:04,469
So when x is equal to 0, y is equal to 1.

58
00:03:04,469 --> 00:03:08,175


59
00:03:08,175 --> 00:03:13,350
So ln of the absolute value of 1 plus 1 squared is equal to

60
00:03:13,349 --> 00:03:16,870
sine of 0 plus c.

61
00:03:16,870 --> 00:03:19,170
The natural log of one, e to the what power is 1?

62
00:03:19,169 --> 00:03:24,530
Well, 0, plus 1 is-- sine of 0 is 0 --is equal to C.

63
00:03:24,530 --> 00:03:27,599
So we get c is equal to 1.

64
00:03:27,599 --> 00:03:32,710
So the solution to this differential equation up here

65
00:03:32,710 --> 00:03:35,740
is, I don't even have to rewrite it, we figured out c

66
00:03:35,740 --> 00:03:37,820
is equal to 1, so we can just scratch this out, and

67
00:03:37,819 --> 00:03:39,019
we could put a 1.

68
00:03:39,020 --> 00:03:42,060
The natural log of the absolute value of y plus y

69
00:03:42,060 --> 00:03:46,210
squared is equal to sine of x plus 1.

70
00:03:46,210 --> 00:03:49,080
And actually, if you were to graph this, you would see that

71
00:03:49,080 --> 00:03:53,430
y never actually dips below or even hits the x-axis.

72
00:03:53,430 --> 00:03:55,469
So you can actually get rid of that absolute

73
00:03:55,469 --> 00:03:56,699
value function there.

74
00:03:56,699 --> 00:03:58,149
But anyway, that's just a little technicality.

75
00:03:58,150 --> 00:04:02,520
But this is the implicit form of the solution to this

76
00:04:02,520 --> 00:04:03,380
differential equation.

77
00:04:03,379 --> 00:04:05,680
That makes sense, because the separable differential

78
00:04:05,680 --> 00:04:06,800
equations are really just

79
00:04:06,800 --> 00:04:09,740
implicit derivatives backwards.

80
00:04:09,740 --> 00:04:13,120
And in general, one thing that's kind of fun about

81
00:04:13,120 --> 00:04:15,810
differential equations, but kind of not as satisfying

82
00:04:15,810 --> 00:04:19,490
about differential equations, is it really is just a whole

83
00:04:19,490 --> 00:04:22,840
hodgepodge of tools to solve different types of equations.

84
00:04:22,839 --> 00:04:27,379
There isn't just one tool or one theory that will solve all

85
00:04:27,379 --> 00:04:28,180
differential equations.

86
00:04:28,180 --> 00:04:30,600
There are few that will solve a certain class of

87
00:04:30,600 --> 00:04:32,810
differential equations, but there's not just one

88
00:04:32,810 --> 00:04:34,319
consistent way to solve all of them.

89
00:04:34,319 --> 00:04:37,099
And even today, there are unsolved differential

90
00:04:37,100 --> 00:04:39,390
equations, where the only way that we know how to get

91
00:04:39,389 --> 00:04:42,659
solutions is using a computer numerically.

92
00:04:42,660 --> 00:04:44,060
And one day I'll do videos on that.

93
00:04:44,060 --> 00:04:47,280
And actually, you'll find that in most applications, that's

94
00:04:47,279 --> 00:04:49,459
what you end up doing anyway, because most differential

95
00:04:49,459 --> 00:04:54,279
equations you encounter in science or with any kind of

96
00:04:54,279 --> 00:04:56,500
science, whether it's economics, or physics, or

97
00:04:56,500 --> 00:05:01,259
engineering, that they often are unsolveable, because they

98
00:05:01,259 --> 00:05:04,050
might have a second or third derivative involved, and

99
00:05:04,050 --> 00:05:05,090
they're going to multiply.

100
00:05:05,089 --> 00:05:06,759
I mean, they're just going to be really complicated, very

101
00:05:06,759 --> 00:05:08,039
hard to solve analytically.

102
00:05:08,040 --> 00:05:10,260
And actually, you are going to solve them numerically, which

103
00:05:10,259 --> 00:05:11,909
is often much easier.

104
00:05:11,910 --> 00:05:14,520
But anyway, hopefully at this point you have a pretty good

105
00:05:14,519 --> 00:05:15,949
sense of separable equations.

106
00:05:15,949 --> 00:05:17,860
They're just implicit differentiation backwards, and

107
00:05:17,860 --> 00:05:20,020
it's really nothing new.

108
00:05:20,019 --> 00:05:24,599
Our next thing we'll learn is exact differential equations,

109
00:05:24,600 --> 00:05:27,210
and then we'll go off into more and more methods.

110
00:05:27,209 --> 00:05:29,639
And then hopefully, by the end of this playlist, you'll have

111
00:05:29,639 --> 00:05:32,469
a nice toolkit of all the different ways to solve at

112
00:05:32,470 --> 00:05:34,890
least the solvable differential equations.

113
00:05:34,889 --> 00:05:36,139
See you in the next video.

114
00:05:36,139 --> 00:05:36,899