1 00:00:00,600 --> 00:00:01,420 ‫Welcome back. 2 00:00:01,890 --> 00:00:09,210 ‫I hope you gave it a thought with regards to the exercise and now let's discuss it, in the first case, 3 00:00:09,690 --> 00:00:16,830 ‫both lambdas must be less than zero, so no one must be less than zero. 4 00:00:17,280 --> 00:00:22,180 ‫And Lunda, too, must be less than zero. 5 00:00:22,950 --> 00:00:23,510 ‫Why? 6 00:00:24,090 --> 00:00:32,130 ‫Well, if you look at the general solution here, then regardless of your A and B constants, negative 7 00:00:32,130 --> 00:00:39,170 ‫lambdas would mean that your Oilor numbers would have negative powers. 8 00:00:40,110 --> 00:00:51,390 ‫So if you have an oil number to the power of minus T or to the power of minus two times T or to the 9 00:00:51,390 --> 00:01:00,840 ‫power of minus two point five times T, whatever lambda you have here for as long as it's negative, 10 00:01:01,200 --> 00:01:11,910 ‫then eight times the oil and number to the power of minus eight times T and let's say that your lambda 11 00:01:11,910 --> 00:01:17,730 ‫one equals minus A and then your B times oil. 12 00:01:17,730 --> 00:01:21,690 ‫And number two, the power of minus B times T. 13 00:01:22,170 --> 00:01:30,120 ‫So your lambda two now equals minus B for as long as your powers are negative. 14 00:01:30,720 --> 00:01:44,040 ‫These things here and here, they will approach to zero as time goes to infinity and they have to the 15 00:01:44,040 --> 00:01:47,970 ‫negative powers will drive the entire thing to zero. 16 00:01:48,870 --> 00:01:54,610 ‫How about if one lambda is less than zero but the other lambda is greater than zero? 17 00:01:55,320 --> 00:02:01,800 ‫Let's say lambda one is less than zero and then lambda two is bigger than zero. 18 00:02:01,830 --> 00:02:02,680 ‫How about that? 19 00:02:02,760 --> 00:02:03,810 ‫What's going to happen then? 20 00:02:04,710 --> 00:02:14,000 ‫Well then this would be false in order to achieve this behavior here in the first graph, because both 21 00:02:14,010 --> 00:02:20,480 ‫numbers must be less than zero if no one is less than zero. 22 00:02:20,820 --> 00:02:28,500 ‫But Lambda two is greater than zero than the first term here eight times. 23 00:02:28,740 --> 00:02:36,750 ‫And then the oilor number to the power of lambda one time stae, this term will approach to zero. 24 00:02:37,350 --> 00:02:41,940 ‫And it is logical because lambda one is less than zero. 25 00:02:42,120 --> 00:02:44,710 ‫So here you will have a negative power. 26 00:02:44,820 --> 00:02:48,960 ‫So this term will approach zero. 27 00:02:48,990 --> 00:02:58,890 ‫Over time, however, the second term B times the oil number to the power of lambda two times T now 28 00:02:59,220 --> 00:03:07,750 ‫this term it has a positive power and therefore this term will simply shoot up. 29 00:03:08,640 --> 00:03:11,430 ‫So this term will approach infinity. 30 00:03:12,370 --> 00:03:22,750 ‫So mathematically, you could write it down like this limit as time goes to infinity, and then here 31 00:03:22,750 --> 00:03:33,550 ‫you have your total error, which is here, and then that equals limit as time goes to infinity, the 32 00:03:33,550 --> 00:03:45,820 ‫first term, which is right here, plus limit as time goes to infinity and the second term, which is 33 00:03:45,820 --> 00:03:48,660 ‫right here, the first term has a negative power. 34 00:03:48,850 --> 00:03:50,960 ‫The second term has a positive power. 35 00:03:51,790 --> 00:03:55,810 ‫So the first term will approach zero. 36 00:03:56,850 --> 00:04:07,580 ‫But the second term will approach infinity, so you will have zero plus infinity equals infinity. 37 00:04:08,310 --> 00:04:13,890 ‫So even if you have one lambda that is negative. 38 00:04:13,900 --> 00:04:22,040 ‫If the other lambda is not negative, then your total error will still go to infinity. 39 00:04:22,920 --> 00:04:29,790 ‫Your first term is not going to be able to hold the second term back because your first term is not 40 00:04:29,790 --> 00:04:31,260 ‫going to become negative. 41 00:04:31,920 --> 00:04:34,280 ‫It will only approach zero. 42 00:04:34,620 --> 00:04:43,080 ‫And since the other one can grow without any boundaries, then this entire error will go to infinity. 43 00:04:43,440 --> 00:04:51,810 ‫As time goes to infinity, even if you're error as a function of time looks like this eight times the 44 00:04:51,810 --> 00:05:02,760 ‫oil and number to the power of minus 1000 times T plus B times the oil number to the power of zero point 45 00:05:03,090 --> 00:05:07,110 ‫zero zero zero one time T. 46 00:05:07,860 --> 00:05:20,010 ‫Then still, one day as time goes to infinity, the first term goes to zero and then the second term 47 00:05:20,550 --> 00:05:24,270 ‫will go to infinity over time. 48 00:05:25,170 --> 00:05:36,600 ‫And so eventually your error as a function of time will still approach zero by the first term and infinity 49 00:05:36,600 --> 00:05:37,860 ‫by the second term. 50 00:05:38,280 --> 00:05:44,220 ‫And so in total, the error will approach infinity over time. 51 00:05:45,210 --> 00:05:55,440 ‫So in the first graph, both lambda one must be less than zero and Lambda two must be less than zero. 52 00:05:55,800 --> 00:05:57,660 ‫And that is true for this graph. 53 00:05:58,170 --> 00:06:08,880 ‫Or even if your graph is like this is the same thing is just here in this graph in the negative region, 54 00:06:09,450 --> 00:06:18,480 ‫you're A and B are negative, or maybe one of them is negative and the other one is positive. 55 00:06:18,900 --> 00:06:22,530 ‫But the one that is negative is more dominant. 56 00:06:23,370 --> 00:06:30,120 ‫But still, if both of your numbers are less than zero, then this entire thing will approach zero no 57 00:06:30,120 --> 00:06:33,200 ‫matter what you're A and B constants are. 58 00:06:34,140 --> 00:06:43,530 ‫So if you want your errors to approach zero, then lambdas are the ones that matter, not A and B constants. 59 00:06:43,530 --> 00:06:48,210 ‫But lambdas in the second graph is the same thing. 60 00:06:48,570 --> 00:06:56,180 ‫But now Lambda one and number two, they must be even more negative to get a stronger response. 61 00:06:56,820 --> 00:07:04,350 ‫So let's say that the numbers in this graph, they have an asterisk here in order to differentiate them 62 00:07:04,830 --> 00:07:07,190 ‫from the lambda in the first graph. 63 00:07:08,070 --> 00:07:14,540 ‫So they have to be more negative than the lambda in the first graph. 64 00:07:15,210 --> 00:07:20,160 ‫And of course, all of them must be less than zero. 65 00:07:21,090 --> 00:07:32,040 ‫And it makes sense, right, if you have an oil no to the power of minus five thousand times time and 66 00:07:32,040 --> 00:07:40,350 ‫then you have another oil and no to the power of minus five times time, then which one will get you 67 00:07:40,350 --> 00:07:42,160 ‫closer to zero faster? 68 00:07:43,020 --> 00:07:53,370 ‫Well, of course, this one here will get you closer to zero faster because its power here is way more 69 00:07:53,370 --> 00:08:04,920 ‫negative than here, minus five times T in the third graph lambda one must be greater than zero or Lunda 70 00:08:04,920 --> 00:08:10,190 ‫two must be greater than zero or both. 71 00:08:11,220 --> 00:08:15,660 ‫So it's in or here or both. 72 00:08:16,740 --> 00:08:24,660 ‫As we discussed earlier, even if one lambda is negative and the other one is positive, then you will 73 00:08:24,660 --> 00:08:27,780 ‫still have this kind of response. 74 00:08:27,790 --> 00:08:35,100 ‫So if you're lambda, one is less than zero and Lambda two is greater than zero, then you will still 75 00:08:35,310 --> 00:08:37,200 ‫have this kind of response. 76 00:08:38,100 --> 00:08:46,290 ‫And the same thing with the fourth graph here, you also need to have at least one lambda that is greater 77 00:08:46,290 --> 00:08:47,020 ‫than zero. 78 00:08:47,190 --> 00:08:56,280 ‫Let's say that your Londa one is greater than zero, but now, of course, your positive lambda must 79 00:08:56,280 --> 00:09:05,700 ‫be even stronger in order to get that stronger rise in the fifth graph you have oscillation and oscillation 80 00:09:05,700 --> 00:09:15,740 ‫means that you have to end up with a cosine or sine terms or both of them. 81 00:09:16,350 --> 00:09:25,010 ‫They need to be present in your solution and that can only happen if your numbers are complex numbers. 82 00:09:25,890 --> 00:09:28,260 ‫So your line does need to be. 83 00:09:29,290 --> 00:09:30,340 ‫Complex. 84 00:09:30,370 --> 00:09:38,710 ‫All right, but since you also wanted to damp out, in other words, you want your error to approach 85 00:09:38,740 --> 00:09:48,430 ‫zero as time approaches infinity, then that means that your lambdas also need to have a negative real 86 00:09:48,430 --> 00:09:48,960 ‫number. 87 00:09:49,570 --> 00:10:03,400 ‫So they look like this number one equals X plus eye times Y, and then number two equals X minus items 88 00:10:03,430 --> 00:10:04,000 ‫Y. 89 00:10:04,900 --> 00:10:08,990 ‫Both of them have the same X, the same real number. 90 00:10:09,400 --> 00:10:16,670 ‫However, if you want this thing to dump out, then that means that your X needs to be less than zero. 91 00:10:17,770 --> 00:10:26,170 ‫So the imaginary part gives you oscillation and then the negative X part gives you error. 92 00:10:26,200 --> 00:10:30,320 ‫That is approaching to zero in the sixth graph. 93 00:10:30,850 --> 00:10:34,090 ‫You also need to have complex Londis. 94 00:10:35,080 --> 00:10:39,900 ‫However, here your real number X must be zero. 95 00:10:40,780 --> 00:10:47,350 ‫Then you are only left with imaginary lambdas, two of them like this. 96 00:10:48,270 --> 00:10:53,340 ‫And then you will only have oscillation, the oscillation will not die out. 97 00:10:54,270 --> 00:11:00,370 ‫And finally in the seventh graph, you also need to have complex lambdas. 98 00:11:00,630 --> 00:11:04,750 ‫However, now your real number must be greater than zero. 99 00:11:05,130 --> 00:11:14,400 ‫Then you will have oscillation, but then your real number will make the error grow in magnitude as 100 00:11:14,400 --> 00:11:14,810 ‫well.