MTH211: CALCULUS: TOPIC: DIFFERENTIAL EQUATIONS USING THE INTEGRATING FACTORS

[00:00:47]Yeah, gentlemen and ladies, you are welcome tonight class.

[00:00:59]Hello, good evening.

[00:01:02]How are you guys?

[00:01:05]How was your day?

[00:01:07]You are welcome once again.

[00:01:10]Okay. Yeah. Tonight uh I will say continue from where we stopped last week, my last class.

[00:01:20]Please like, make sure you share and uh also invite others.

[00:01:27]Uh we solved one example. We are going to solve another example.

[00:01:34]The example says determine the general solution of x dx = 2 - 4x² is simple as ABC. We are still using separation of variables to um to determine the first order differential equation.

[00:02:00]After that we will still use what is called integrating factor to uh determine the first order differential equation. It's simple as ABC is that clear? So let's uh proceed.

[00:02:15]All right solution right our x uh dy all over dx is = 2 - 4x².

[00:02:28]All right. So now how can we separate the variables? We have x here dy / dx = 2 - 4x². So who can tell us and what to do? Look at the board.

[00:02:45]Who can tell us and what to do?

[00:02:49]Yeah.

[00:02:52]And nobody is okay. So if you can see what I am saying or what you will do first is to divide through by this x. So divide uh through by x. So when you divide 2 by x this is what you have x d y dx all x = 2 - 4x² all over x two of us. So this and this will go be left with dy all over dx = to I want us to take note of something here. Now if you can see what I'm saying when we have a + b all over x this is the same as a all x + b all x two of us.

[00:03:50]So if you agree with me it means here we are going to have f_sub_2 all x - 4x² all over x right so this and this will cancel to give us the y all over dx = 2 all x - 4x two of us any question at this point please do you have any question Any question at this point?

[00:04:25]Do you have any question at this point?

[00:04:28]Please ask your question. I'll be very glad to answer your question.

[00:04:36]Do you have any question?

[00:04:39]Yes or no?

[00:04:42]Oh, so we can proceed, right? So you understand what is going on here. So what else we will cross multiply? So we are going to have dy = to open bracket 2 all / x - 4x dx. So at this point we integrate integrating integrating both side right.

[00:05:12]Okay. So when we integrate both side this is what we'll have integral open bracket uh uh 2 all x - 4x dx two of course okay uh if you can see what I am saying you don't need a prophet to tell you so here look at the board if you integrate dy you are supposed to have y Then here try to open the bracket with the integral dx. So here will be integral 2 all / x dx. Then - 4x dx. Three of us. Let me write it like a mathematician.

[00:06:00]Okay. Three of us. At this point, do you have any question? Please ask. Any question.

[00:06:09]Any question? Ask your question please. Ask ask your question. Any question? Yes or no?

[00:06:30]Okay. It is there right?

[00:06:44]No question. Okay. Is okay. Let's proceed if there's no question. So I want us to note that uh when we have uh when we have uh something of this form 2 all x this is the same thing as uh 2 * 1 all x two of us. So we can write these two outside the integral 1 all x dx minus write the form outside the integral x dx because they are constants two of course. So y will be equals to when we integrate uh 1 all x we are supposed to have e x then what do we have here two? So 2 in x integrate minus integrate x we have for here right 4 * x^ 1 + 1 all 1 + 1 + any constant which is k. So here will be y = 2 in x - 4x² all over 2 + uh k. Then this is the same as y = 2 in x - what? 2x² + uh k two of us. This is what you are supposed to have.

[00:08:23]Do you have any question? [groaning] Any question?

[00:08:34]Yes. Buari Sani ask your question.

[00:08:41]Buhari you want to ask question please ask [groaning] so that we can proceed uh so that we can proceed.

[00:08:59]No question.

[00:09:02][groaning] No question is okay. Let's proceed further.

[00:09:12]The integration part sir how sir throw more light is okay.

[00:09:21]uh I know very well that some of you is your first time of meeting us here. So unfortunate uh you know integration is let us assume this is a function a x^ of n dx right here when integrate x^ n dx it means this will be a x ra to the power of n + 1 all over plus n + 1 + any constant either k or b.

[00:10:07]So can you see now this is how to integrate any given function. Oh.

[00:10:13]Oh this is uh sorry about it. They just to do the lights.

[00:10:25]He lost to the lights.

[00:10:28][groaning] But hold on.

[00:10:36]Hold on. If they bring the lights, I will be I will come back.

[00:10:44]I will come back.