1 00:00:00,610 --> 00:00:09,520 ‫And now you might think that, wait a minute, Setu had an imaginary number in it, it had this eye 2 00:00:09,550 --> 00:00:10,670 ‫in it, right. 3 00:00:11,590 --> 00:00:23,320 ‫So this was your CI one and this is your C two and C one was one, like you can see here. 4 00:00:23,860 --> 00:00:28,330 ‫And then C two ended up being minus two. 5 00:00:28,720 --> 00:00:30,130 ‫Over three. 6 00:00:31,210 --> 00:00:43,510 ‫So how can see to be a real number, minus two, over three, if it's also I times A minus B. 7 00:00:44,580 --> 00:00:51,960 ‫You might think that it's a complex number because it has an eye here, but it turns out to be a real 8 00:00:51,960 --> 00:00:52,450 ‫number. 9 00:00:52,470 --> 00:00:53,620 ‫So how can it be? 10 00:00:54,510 --> 00:01:04,130 ‫Well, that just means that the A or B, Constance or both have also an imaginary part in them. 11 00:01:04,500 --> 00:01:09,770 ‫And if you put them all together, then the imaginary parts cancel out. 12 00:01:10,140 --> 00:01:11,050 ‫Let's see that. 13 00:01:11,940 --> 00:01:18,450 ‫So you know that A plus B equals one because that's your C one. 14 00:01:18,600 --> 00:01:19,080 ‫Right. 15 00:01:19,380 --> 00:01:23,580 ‫And that means that you can rewrite it like this. 16 00:01:24,300 --> 00:01:31,920 ‫A equals one minus B, then C two was eight times. 17 00:01:31,930 --> 00:01:40,830 ‫And in the parentheses you will have A minus B and that equals minus two. 18 00:01:40,890 --> 00:01:41,830 ‫Over three. 19 00:01:42,630 --> 00:01:52,050 ‫And now what I'm going to do instead of this A, I'm going to put here one minus B like this and then 20 00:01:52,170 --> 00:01:53,920 ‫another B here. 21 00:01:54,960 --> 00:02:08,360 ‫So what you have here is I times one minus to be in that equals minus two over three. 22 00:02:09,330 --> 00:02:11,880 ‫So I can write it down like this. 23 00:02:12,280 --> 00:02:24,300 ‫I'm going to multiply by one and by minus two times B, so I'm going to have I minus two times I times 24 00:02:24,510 --> 00:02:29,150 ‫B equals minus two, over three. 25 00:02:29,970 --> 00:02:39,840 ‫I can rewrite this entire thing like this I plus two over three because I'm going to put this minus 26 00:02:40,110 --> 00:02:52,140 ‫two over three on the other side of the equation sine and so all that equals two times I times B and 27 00:02:52,140 --> 00:03:05,190 ‫that means that B equals I plus two over A three and then all that divided by two times I. 28 00:03:05,850 --> 00:03:09,570 ‫I can also rewrite B like this. 29 00:03:10,140 --> 00:03:25,590 ‫So you're going to have I divided by two times I and then plus two over and then two times I times three 30 00:03:26,310 --> 00:03:27,000 ‫like this. 31 00:03:27,870 --> 00:03:30,710 ‫So I can just rewrite this form in this form. 32 00:03:31,470 --> 00:03:37,200 ‫I can cancel out the eyes here and then I can cancel out the two here. 33 00:03:37,950 --> 00:03:56,400 ‫And so B equals one half plus one over three times I and since A equals one minus B that means that 34 00:03:56,400 --> 00:04:00,120 ‫it equals one minus. 35 00:04:00,270 --> 00:04:12,090 ‫And then you have here one half plus one over three times I and so we can rewrite a like this one minus 36 00:04:12,090 --> 00:04:21,750 ‫one half is going to be one half and then minus one over three times I. 37 00:04:22,320 --> 00:04:24,700 ‫And so that is your a. 38 00:04:25,440 --> 00:04:37,280 ‫And now let's do the reverse C one equals A plus B, you can see it from here. 39 00:04:37,740 --> 00:04:47,460 ‫And so you're A is one half minus one over three times I. 40 00:04:48,460 --> 00:05:02,770 ‫And then plus B, your B is plus one 1/2 plus one over three times I, so you can see that you can get 41 00:05:02,770 --> 00:05:07,230 ‫rid of this term and this term because they amount to zero. 42 00:05:07,900 --> 00:05:19,840 ‫And then here you have one half plus one half, which equals one and then C two equals I times A minus 43 00:05:19,840 --> 00:05:34,900 ‫B. In other words I times A was one half minus one over three times I and now you have a minus sign 44 00:05:34,900 --> 00:05:41,830 ‫here and your B was one 1/2 plus one over three times I. 45 00:05:41,980 --> 00:05:53,570 ‫So because of this minus sign you will have here minus one half, minus one over three times. 46 00:05:53,590 --> 00:05:55,870 ‫I like this. 47 00:05:56,380 --> 00:06:04,120 ‫And now you can get rid of one half and minus one half because they amount to zero and so you will have 48 00:06:04,120 --> 00:06:15,730 ‫eight times and then if you add these two terms together you will have minus two over three times I 49 00:06:16,540 --> 00:06:20,920 ‫you can cancel out the eyes and then you will have minus two. 50 00:06:21,100 --> 00:06:22,380 ‫Over three. 51 00:06:23,110 --> 00:06:24,720 ‫That was your C two. 52 00:06:25,810 --> 00:06:36,490 ‫So you see the imaginary numbers are hidden inside the A and B constants in such a way that in total 53 00:06:36,670 --> 00:06:42,340 ‫the imaginary parts cancel out and you're left with a real number constant. 54 00:06:43,210 --> 00:06:46,030 ‫So you'll C one and see two. 55 00:06:46,030 --> 00:06:48,580 ‫They're real numbers in the end. 56 00:06:49,420 --> 00:06:56,140 ‫And by the way, when you have the second order differential equation and you have complex routes, 57 00:06:56,620 --> 00:07:07,240 ‫then those complex routes are always like this X plus minus and then you will have eight times Y, so 58 00:07:07,240 --> 00:07:08,740 ‫you have to lend us. 59 00:07:09,190 --> 00:07:20,800 ‫But both of those lenders have the same real number and then one of them has a plus eight times y imaginary 60 00:07:20,800 --> 00:07:27,420 ‫part and then the other lambda has minus eight times Y imaginary part. 61 00:07:28,420 --> 00:07:30,240 ‫So they are always conjugates. 62 00:07:31,150 --> 00:07:43,780 ‫So in our case it was two plus minus and then I times three you see you have the same real part, but 63 00:07:43,780 --> 00:07:53,410 ‫then you have plus I times three imaginary part for one lambda and then minus I times three for another 64 00:07:53,410 --> 00:08:08,290 ‫lambda lambda one equals this and lambda two equals this right here and now as an exercise, take this 65 00:08:08,530 --> 00:08:18,880 ‫solution that is particular to our initial conditions and check if it really is the right solution for 66 00:08:18,880 --> 00:08:21,340 ‫this differential equation here. 67 00:08:21,790 --> 00:08:28,720 ‫For this one again, take the first derivative of it and then take the second derivative of it and then 68 00:08:29,470 --> 00:08:37,210 ‫put them inside your differential equation, including the solution itself, and see if zero actually 69 00:08:37,210 --> 00:08:38,360 ‫equals zero. 70 00:08:39,040 --> 00:08:47,580 ‫Try doing it exactly like before and see if you can check if the solution is truly correct. 71 00:08:48,670 --> 00:08:52,030 ‫So thank you very much and I'll see you in the next video.