1 00:00:00,470 --> 00:00:01,320 ‫Welcome back. 2 00:00:01,910 --> 00:00:09,800 ‫Let's now perform the solution check, so this is our solution for this differential equation, while 3 00:00:09,800 --> 00:00:19,640 ‫having the initial conditions like this error at time equals zero seconds equals one meter and then 4 00:00:19,640 --> 00:00:21,230 ‫error dot the time. 5 00:00:21,230 --> 00:00:29,110 ‫Derivative of the error at time equals zero seconds equals zero meters per second. 6 00:00:29,990 --> 00:00:33,350 ‫So let's not take the derivative of this solution. 7 00:00:34,340 --> 00:00:37,120 ‫This derivative looks like this. 8 00:00:37,790 --> 00:00:45,080 ‫First of all, you take the derivative of this part and you leave the other part alone and then you 9 00:00:45,080 --> 00:00:46,250 ‫do it in reverse. 10 00:00:46,760 --> 00:00:53,390 ‫The oil, the number to the power of two times, you leave it alone and then you take the derivative 11 00:00:53,390 --> 00:00:56,990 ‫of what you have in the brackets here. 12 00:00:57,710 --> 00:01:01,220 ‫And so here you have three times two over three. 13 00:01:01,250 --> 00:01:09,680 ‫You can cancel out the threes, then you can rewrite this entire thing like this where you multiply 14 00:01:09,680 --> 00:01:18,400 ‫two by what you have here in this bracket, and then you will have here minus four over three. 15 00:01:19,190 --> 00:01:20,630 ‫And here you will have two. 16 00:01:21,440 --> 00:01:31,070 ‫And you can see that two times cosine, three times t can be cancelled out with this minus two times 17 00:01:31,070 --> 00:01:33,950 ‫cosine three times T, right. 18 00:01:33,970 --> 00:01:37,910 ‫So this one and this one, they will be canceled out. 19 00:01:38,600 --> 00:01:51,950 ‫And so your error dot equals minus thirteen over three times the oil number to the power of two times 20 00:01:51,950 --> 00:01:59,810 ‫T and then times sine and then you have three times T here. 21 00:02:00,590 --> 00:02:12,500 ‫So that's your solution derivative then this will be the second derivative of your error and now you're 22 00:02:12,500 --> 00:02:16,370 ‫going to put all these things into your differential equation here. 23 00:02:17,360 --> 00:02:19,910 ‫This will be your second derivative here. 24 00:02:19,910 --> 00:02:31,610 ‫So e double that that you put here, then this here, this one will be your first derivative, which 25 00:02:31,610 --> 00:02:35,690 ‫is this one here that you're going to put right here. 26 00:02:36,290 --> 00:02:44,690 ‫And then you have minus four times this error that so you see you have here minus four times the error 27 00:02:44,710 --> 00:02:48,950 ‫dot here and then this one here. 28 00:02:49,310 --> 00:02:53,570 ‫This will be your 13 times error. 29 00:02:54,200 --> 00:02:56,830 ‫Your error solution is this one here. 30 00:02:57,740 --> 00:03:05,620 ‫And so you can see that you can cancel out this term with this term here because they amount to zero. 31 00:03:06,470 --> 00:03:18,650 ‫And so if you put it all together, then you will have minus twenty six over three plus fifty two over 32 00:03:18,800 --> 00:03:30,080 ‫three and then minus twenty six over three and all that multiplied by the oil, a number to the power 33 00:03:30,080 --> 00:03:44,210 ‫of two times T times sine and then three T, so minus twenty six over three comes from here and then 34 00:03:44,390 --> 00:03:47,480 ‫here you will have fifty to over three. 35 00:03:47,630 --> 00:03:55,100 ‫If you multiply minus four times minus thirteen over three then you will get fifty two of three and 36 00:03:55,100 --> 00:04:02,630 ‫then here you will have minus 13 times two over three and you will get minus twenty six over three. 37 00:04:03,290 --> 00:04:12,350 ‫And then you will factor out this oil number with its power and sine three times t you will factor them 38 00:04:12,350 --> 00:04:19,240 ‫out and you can see that here in this bracket you will have zero. 39 00:04:20,000 --> 00:04:25,780 ‫So if you multiply zero by all this here, then you will get zero. 40 00:04:26,450 --> 00:04:30,050 ‫And look, this part here will be zero. 41 00:04:30,800 --> 00:04:34,660 ‫And then on this side of the equation, sine you will also have zero. 42 00:04:35,000 --> 00:04:42,740 ‫So zero equals zero and therefore the solution is correct. 43 00:04:43,440 --> 00:04:50,950 ‫Alright then in the next video, we're going to go back to our controller context. 44 00:04:51,320 --> 00:04:52,250 ‫Thank you very much.