1
00:00:00,420 --> 00:00:09,960
‫And so if this is your general solution here and then you have these kind of relationships here, then

2
00:00:10,890 --> 00:00:22,190
‫you can rewrite our general solution like this, the error as a function of time equals and then a times

3
00:00:22,230 --> 00:00:24,960
‫the oil, the no to the power of two times T.

4
00:00:25,740 --> 00:00:38,970
‫And now instead of this, I'm going to write it like this callsign three times T plus I times sine three

5
00:00:38,970 --> 00:00:41,460
‫times T like this.

6
00:00:42,270 --> 00:00:43,200
‫And I can do that.

7
00:00:43,220 --> 00:00:50,980
‫I'm just following this formula here and then the other term will be like this plus B times the oil,

8
00:00:50,980 --> 00:01:00,600
‫the number to the power of two times T and now instead of this I'm going to write it like this times.

9
00:01:00,600 --> 00:01:14,010
‫And then in the brackets you will have cosine three times T minus I times sine three times T like this.

10
00:01:15,000 --> 00:01:22,410
‫And now I'm going to factor out the oil and number to the power of two times T and then in the brackets

11
00:01:22,410 --> 00:01:35,910
‫I will have eight times cosine, three times T plus eight times I times sine three times T plus B times

12
00:01:35,910 --> 00:01:49,170
‫cosine three times T and now minus B times I times sine three times T like this.

13
00:01:50,040 --> 00:01:56,310
‫And so inside the brackets I can factor out the cosigns here.

14
00:01:57,300 --> 00:02:08,310
‫So I'm going to put A plus B here, times callsign three times T, and now I'm going to factor out the

15
00:02:08,310 --> 00:02:10,400
‫signs here as well.

16
00:02:10,710 --> 00:02:23,010
‫So I'm going to have A minus B times I times sine three times T like this.

17
00:02:24,060 --> 00:02:32,430
‫And now I'm going to say that this is A plus B, I'm going to just say that this entire thing is a more

18
00:02:32,430 --> 00:02:35,220
‫general constant C one.

19
00:02:35,820 --> 00:02:42,420
‫And then here I'm going to take this entire thing, including the imaginary eye number, and I'm going

20
00:02:42,420 --> 00:02:44,810
‫to say that it's C two.

21
00:02:44,850 --> 00:02:47,640
‫So just another random constant.

22
00:02:47,850 --> 00:02:53,220
‫So C one and C two, they are random constants, just like A plus B.

23
00:02:54,440 --> 00:03:01,670
‫So the general solution is error as a function of time, and then you have this or the number to the

24
00:03:01,670 --> 00:03:14,480
‫power of two times T and then inside the brackets you will have C one times cosine, three times T and

25
00:03:14,480 --> 00:03:24,010
‫then plus C, two times sine and then three times T here.

26
00:03:24,020 --> 00:03:25,800
‫So that's your general solution.

27
00:03:26,750 --> 00:03:38,570
‫And now to find C one and C two, let's again assume that the initial conditions are like this error

28
00:03:38,600 --> 00:03:51,280
‫at time equals zero equals one meter and the error dot at time equals zero equals zero meters per second.

29
00:03:52,100 --> 00:03:55,510
‫So let's take the derivative of this solution here.

30
00:03:56,510 --> 00:04:00,590
‫So error dot is a function of time equals.

31
00:04:00,950 --> 00:04:06,710
‫And now according to the product rule from calculus, we first take the derivative of this part.

32
00:04:07,200 --> 00:04:15,200
‫So you're going to have two times the oil and number to the power of two times T and then what you have

33
00:04:15,200 --> 00:04:15,950
‫in the brackets.

34
00:04:15,950 --> 00:04:17,270
‫We don't change that.

35
00:04:18,140 --> 00:04:26,840
‫So it's going to be like this and now we're going to leave this unchanged, the oil and no to the power

36
00:04:26,840 --> 00:04:33,980
‫of two times T and we take the derivative of the functions in the brackets.

37
00:04:34,460 --> 00:04:51,890
‫So you will have minus three C one times sine three times T and then plus three times C two and then

38
00:04:52,280 --> 00:04:57,050
‫cosine three times T like this.

39
00:04:57,980 --> 00:05:05,210
‫Here you have the threes because of the chain rule, because you also need to take the derivative inside

40
00:05:05,210 --> 00:05:07,820
‫the cosine and sine parentheses.

41
00:05:08,880 --> 00:05:18,750
‫And so according to our initial conditions here, error at time equals zero seconds, then all that

42
00:05:18,750 --> 00:05:21,240
‫equals one meter.

43
00:05:21,270 --> 00:05:24,510
‫So we're going to put one here and here.

44
00:05:25,080 --> 00:05:32,190
‫Error dot time equals zero seconds, and then all that equals a zero.

45
00:05:33,310 --> 00:05:42,190
‫Now, if time equals zero, then this thing here becomes one, because the oil number to the power of

46
00:05:42,190 --> 00:05:45,310
‫two times zero equals the oil.

47
00:05:45,310 --> 00:05:48,460
‫The number to the power of zero equals one.

48
00:05:49,400 --> 00:05:59,080
‫And then this cool three time city will also be won and then signed three times T one time equals zero,

49
00:05:59,600 --> 00:06:10,700
‫then it will be zero and then in the error derivative function, again, this part will be one, then

50
00:06:11,030 --> 00:06:18,270
‫the cosine part will be one, then the sine part will be zero.

51
00:06:18,950 --> 00:06:21,770
‫Again, this part here will be one.

52
00:06:22,430 --> 00:06:25,190
‫The sign part here will be zero.

53
00:06:25,190 --> 00:06:27,860
‫And then the coastline part here will be one.

54
00:06:28,910 --> 00:06:34,480
‫So the error at time equals zero seconds equals.

55
00:06:34,910 --> 00:06:43,220
‫And from here you can see that since this entire term will also be zero because you will be multiplying

56
00:06:43,340 --> 00:06:45,380
‫C two times zero.

57
00:06:46,370 --> 00:06:56,090
‫That means that you will have it like this error at time equals zero seconds equals C one, and that's

58
00:06:56,090 --> 00:06:56,380
‫it.

59
00:06:56,390 --> 00:06:58,790
‫That equals your one, right?

60
00:06:58,820 --> 00:07:03,950
‫So here you will only have your C one here and then you equate that to one.

61
00:07:04,190 --> 00:07:07,670
‫So you know that your C one equals one.

62
00:07:07,700 --> 00:07:09,980
‫You have one of your constants.

63
00:07:10,920 --> 00:07:23,070
‫And then your error dot at time equals zero seconds equals from the first term, you will have two times

64
00:07:23,580 --> 00:07:32,460
‫see one and from the second term you will have plus three times C two.

65
00:07:32,940 --> 00:07:35,280
‫And all that will equal to zero.

66
00:07:36,210 --> 00:07:41,090
‫Since you already know that your C one equals one, you can just put one here.

67
00:07:41,460 --> 00:07:50,080
‫So you will have two times one plus three times C, two equals zero.

68
00:07:51,000 --> 00:07:58,370
‫And now to find C two, you put two on the other side of the equation sine and you divide it by three.

69
00:07:58,950 --> 00:08:13,140
‫So three times C two equals minus two and well the three goes here so C two equals minus two over three.

70
00:08:14,280 --> 00:08:24,090
‫So the solution for the given initial values is the following error as a function of time equals the

71
00:08:24,090 --> 00:08:35,370
‫oil number to the power of two times t like here and now in the brackets you will have it like this

72
00:08:35,730 --> 00:08:37,320
‫C one here.

73
00:08:37,320 --> 00:08:38,630
‫That was your one.

74
00:08:38,640 --> 00:08:50,730
‫So one times cosine, three times T and now your C two was minus two of a three.

75
00:08:50,760 --> 00:08:59,840
‫So you will have it here minus two over three times sine and then three times T.

76
00:09:00,630 --> 00:09:07,680
‫So that will be your solution for these particular initial conditions.