1 00:00:00,540 --> 00:00:01,350 ‫Welcome back. 2 00:00:02,010 --> 00:00:09,120 ‫So now we are going to derive a relationship between the polls and the key values in our differential 3 00:00:09,120 --> 00:00:22,860 ‫equation that looked like this error, double that minus K, two times error, Dott minus K one times 4 00:00:23,160 --> 00:00:34,070 ‫the error equals zero, essentially by choosing the polls that make the error behave like we want to, 5 00:00:34,470 --> 00:00:42,980 ‫and then by calculating the key values, we construct a differential equation suitable for us. 6 00:00:43,890 --> 00:00:56,340 ‫So our general solution is error as a function of time equals and then a random constancy and then the 7 00:00:56,340 --> 00:01:04,620 ‫oil number to the power of Lambda Times t this is our general solution. 8 00:01:05,220 --> 00:01:13,500 ‫Then we take the derivative of it and we will have error dot as a function of time equals and then C 9 00:01:13,710 --> 00:01:25,140 ‫times lambda times the oil or no to the power of Lambda Times T and then after we have taken the first 10 00:01:25,140 --> 00:01:29,070 ‫derivative of it, we're going to take the second derivative of it. 11 00:01:29,340 --> 00:01:40,110 ‫So e double that as a function of time equals C times lambda squared times the oilor number to the power 12 00:01:40,110 --> 00:01:47,820 ‫of Lambda Times T and now we're going to put all those three things into our differential equation. 13 00:01:48,540 --> 00:01:52,590 ‫First of all, the error double dot is this one here. 14 00:01:52,590 --> 00:02:04,500 ‫So C times lambda squared times the oil and number to the power of Lambda Times T then minus K two that 15 00:02:04,500 --> 00:02:08,440 ‫you have here times the error dot which is this one here. 16 00:02:08,880 --> 00:02:20,940 ‫So C times lambda times the oil number to the power of Lambda Time C and then finally minus K one that 17 00:02:20,940 --> 00:02:24,600 ‫you have here and then you multiplied by the error itself. 18 00:02:24,780 --> 00:02:32,760 ‫So times C times the oil, the number to the power of Lambda Times T and all that of course equals to 19 00:02:32,760 --> 00:02:33,190 ‫zero. 20 00:02:34,020 --> 00:02:45,300 ‫You can factor out C and also the oil, a number to the power of the time D and then you will have in 21 00:02:45,300 --> 00:02:55,920 ‫the brackets this expression lambda squared minus K two times lambda minus K one. 22 00:02:56,490 --> 00:03:05,430 ‫And all that equals to zero, assuming that the constancy cannot equal to zero and also the oil and 23 00:03:05,430 --> 00:03:07,740 ‫number to the power of the time. 24 00:03:07,740 --> 00:03:10,280 ‫C cannot be zero either. 25 00:03:11,040 --> 00:03:18,070 ‫That means that this equation can only be zero if what you have in the brackets is zero. 26 00:03:18,870 --> 00:03:28,630 ‫So lambda squared minus K two times lambda minus K one that must equal to zero. 27 00:03:29,340 --> 00:03:33,040 ‫And now we get the lambda or the roots or the polls. 28 00:03:33,570 --> 00:03:38,640 ‫So you see you have three names for the same thing. 29 00:03:39,060 --> 00:03:44,460 ‫Route's polls and lambdas and power constants. 30 00:03:45,240 --> 00:03:47,100 ‫Eventually it's the same thing. 31 00:03:47,700 --> 00:03:51,030 ‫So lambda one and two equals. 32 00:03:51,630 --> 00:03:59,370 ‫And then according to the formula, in order to find the roots for this quadratic equation, you have 33 00:03:59,880 --> 00:04:15,270 ‫K two divided by two, then plus minus and then square root and then K two over to all that is taken 34 00:04:15,270 --> 00:04:16,800 ‫to the power of two. 35 00:04:16,950 --> 00:04:23,540 ‫So you square it and then you're going to have plus K one here. 36 00:04:24,330 --> 00:04:30,560 ‫So that will be your lambda one and that will be your lambda too. 37 00:04:31,410 --> 00:04:40,170 ‫So you have expressed your lambdas in terms of case you have lambda one as a function of K one and K 38 00:04:40,170 --> 00:04:49,390 ‫to and then you have lambda two as a function of K one and K two and now you will have an exercise. 39 00:04:49,980 --> 00:04:54,180 ‫Now you simply have to express the case in terms of Londis. 40 00:04:55,230 --> 00:04:59,280 ‫So you need to have an expression for K one as a function of. 41 00:04:59,840 --> 00:05:07,130 ‫The one and Lunda two, and then you need to have an expression for K two as a function of Lunda one 42 00:05:07,490 --> 00:05:09,080 ‫and Lunda to. 43 00:05:09,960 --> 00:05:21,000 ‫And so your exercise now is to rearrange these equations in order to get this and this right now you 44 00:05:21,000 --> 00:05:28,970 ‫have this and this, but you need to rearrange these equations to get this and this. 45 00:05:29,040 --> 00:05:30,520 ‫That will be your exercise. 46 00:05:31,590 --> 00:05:34,680 ‫So give it a try and we'll talk about it in the next video. 47 00:05:34,980 --> 00:05:35,850 ‫Thank you very much.