1 00:00:00,570 --> 00:00:01,380 ‫Welcome back. 2 00:00:01,860 --> 00:00:04,620 ‫So let's now check our solutions. 3 00:00:05,010 --> 00:00:09,770 ‫This is your solution for this differential equation. 4 00:00:10,200 --> 00:00:19,080 ‫And remember, these concerns are three or five and two or five, because your error at time equals 5 00:00:19,320 --> 00:00:21,920 ‫zero equals one. 6 00:00:21,930 --> 00:00:28,510 ‫And then the time derivative of your error at time equals zero equals zero. 7 00:00:28,860 --> 00:00:36,510 ‫So the error was in meters in our case, because we have X, Y and Z dimensions and they are measured 8 00:00:36,510 --> 00:00:37,290 ‫in meters. 9 00:00:38,070 --> 00:00:45,990 ‫And then the error that was meters per second because you have a time derivative here. 10 00:00:46,800 --> 00:00:55,920 ‫And so if we take the derivative of the error of this general solution, then you will have it like 11 00:00:55,920 --> 00:01:08,820 ‫this three over five times two, and then the oil and number to the power of two times T plus and then 12 00:01:09,240 --> 00:01:14,790 ‫two over five and then times minus three. 13 00:01:15,330 --> 00:01:18,630 ‫And then the oil, the number to the power of minus three times. 14 00:01:19,740 --> 00:01:26,200 ‫In other words, it equals six over five times the oil. 15 00:01:26,260 --> 00:01:35,640 ‫Number to the power of two times Steet and then minus six over five times the oil number to the power 16 00:01:35,640 --> 00:01:45,950 ‫of minus three times T and well, the second time derivative of this general solution will be like this. 17 00:01:46,590 --> 00:01:59,280 ‫So six over five here, times two and then again times the oil number to the power of two times T and 18 00:01:59,280 --> 00:02:13,770 ‫then minus six over five like here times minus three times the oil or no to the power of minus three 19 00:02:14,310 --> 00:02:20,580 ‫times T and so you're going to have 12 over five. 20 00:02:21,500 --> 00:02:33,380 ‫And then the oil, the number to its power, and then plus 18 over five times the oil, the number, 21 00:02:33,380 --> 00:02:47,360 ‫and again, it's another power and now this thing here will go here, then this thing here will go here. 22 00:02:47,750 --> 00:02:53,800 ‫And then this thing here that will go here. 23 00:02:54,770 --> 00:02:58,700 ‫That would be the second time derivative for this one here. 24 00:02:59,650 --> 00:03:09,400 ‫That would be the first time derivative for this one here and then minus six times and then the expression 25 00:03:09,400 --> 00:03:11,260 ‫for the error itself. 26 00:03:12,130 --> 00:03:16,930 ‫And so if you write it all out, then you will have it in this form here. 27 00:03:17,800 --> 00:03:27,610 ‫And then you can factor out the oil number to the power of two times T and then in the brackets you 28 00:03:27,610 --> 00:03:28,660 ‫will have this thing. 29 00:03:28,870 --> 00:03:37,360 ‫And then you can also factor out the oil, a number to the power of minus three times T, and this is 30 00:03:37,360 --> 00:03:39,070 ‫what you will have in the brackets. 31 00:03:39,550 --> 00:03:48,440 ‫And so that entire thing has to equal to zero because this differential equation here equals to zero. 32 00:03:48,880 --> 00:03:55,050 ‫So on the right side of the equation side, you have a zero here and it's the same zero like here. 33 00:03:55,450 --> 00:03:59,560 ‫So this entire thing has to equal zero. 34 00:04:00,250 --> 00:04:10,520 ‫And it does because in the brackets here, this equals zero and also this equals zero. 35 00:04:11,440 --> 00:04:14,860 ‫So the left side of the equation sine will also be zero. 36 00:04:15,580 --> 00:04:18,850 ‫And so zero equals zero. 37 00:04:18,850 --> 00:04:24,900 ‫And therefore your solution, which is this one here, is correct. 38 00:04:25,780 --> 00:04:29,530 ‫And so that's how you check whether your solution is correct or not. 39 00:04:29,860 --> 00:04:35,830 ‫You take this function, you take the first derivative, you take the second derivative, and you put 40 00:04:35,830 --> 00:04:40,090 ‫all that inside your original differential equation. 41 00:04:40,090 --> 00:04:47,080 ‫And you see if the number on one side of the equation sine is the same number, like on the other side 42 00:04:47,080 --> 00:04:48,160 ‫of the equation sine. 43 00:04:48,760 --> 00:04:50,040 ‫And that's it for that. 44 00:04:50,920 --> 00:04:53,770 ‫But now I'm going to give you another exercise. 45 00:04:54,400 --> 00:05:01,690 ‫If you remember, our very original differential equation was error double date minus. 46 00:05:02,710 --> 00:05:10,830 ‫OK, two times ever that minus K, one times error equals zero. 47 00:05:11,900 --> 00:05:21,830 ‫And let's say that now your key values are like this, your key one equals minus 13. 48 00:05:22,070 --> 00:05:32,910 ‫All right, so this key, one value, that's this one here, and then your key to value equals four. 49 00:05:33,770 --> 00:05:37,070 ‫So this key to value, that's this one here. 50 00:05:37,730 --> 00:05:47,210 ‫And so your differential equation now is erodable dot minus and then K2 is four. 51 00:05:47,210 --> 00:05:54,310 ‫So four times error dot and then minus and then K one is minus 13. 52 00:05:54,800 --> 00:06:00,410 ‫So you will end up having plus 13 here times. 53 00:06:00,410 --> 00:06:02,240 ‫Error equals zero. 54 00:06:03,080 --> 00:06:07,670 ‫And your exercise now is to solve this differential equation. 55 00:06:08,090 --> 00:06:12,080 ‫Start doing it just like we did with the previous differential equation. 56 00:06:12,500 --> 00:06:17,330 ‫And don't worry, you will see in the end how it all comes together in our controller. 57 00:06:17,810 --> 00:06:20,620 ‫I'm just laying the groundwork for that. 58 00:06:21,080 --> 00:06:25,730 ‫So try to solve this differential equation and I'll see you in the next video. 59 00:06:26,030 --> 00:06:26,930 ‫Thank you very much.