Okay, let's get Turk. So s so far, so far happier. I you have learned to solve the second order linear, different occasion. However, in this case, we have nonlinear differential occasion because of peace. Sign terms. So what's your first? Just tap. You have legalized thiss differential. Kishan. Right? So if you look at the hint the interest. As for a small values of Sarah, we can use to linear approximation. And so his science era is almost the same. It is Stella. So we can transform this differential equation right now. Uh, look, att Now let's move on. Lookinto, eh? Find the occasional motion of a pendulum with lengths. A one meter, Which means he said l is one. And then ah, Farah is initially pointed to radium. That means Sarah, zero is point two and then Ah, initial angular velocity is one. Right? So we have thes second order different, alina, different education. Right? So if you have ah, linear different occasion, what is your first step? Your first stubby is so the characters peopling amir. So this represents the characters to pull in. We're right. So in this case, it's a conductor. So esseker plus G you're here and she's a constant Ichi's thie Accreditation. Acceleration. Right. So if you served these characters Polina, Mira, you get too complex. Ask complex delusions. Right? So if you have a complex solutions, then what is your solution? What is your real solution? What's your solution? Off the solution to different occasion. The solution has to form or decent, right? It consists of Lisa Linear linear combination, off sign and consign. Right. And here a and B are just constant. So what is your next channel? Way? Have to find these A and B. How are using these two initial conditions? Right. So when he questions zero the Asia and Angle Wass longitude or Indian and the initial angular velocity US one. Right. So by using these two conditions, we can sew way confined to constant A and B. So if we plug zero two key, then it becomes hero. So your piece going to two and then if you differentiate it and then plug zero, then this term will vanish. So you get scare ittle g times A, of course, one which means a course one over Skerritt of G. Yeah, right. So he's easier. Finer answer to a I hear it. It is two point two, right? So let's move on. Be what is the maximum angle from the vertical? So we have to find maximum destruction. Great. And then there can be two ways. Where is it used? D'Oh Ah, extreme value deuterium. So, by finding the solution of the situation right to remember this one Yeah. By finding the solution ofthis occasion, you can find the local extreme a right and then by using using the second derivative test you can test, you can determine whether there is a local maximum or local minimum. Then you can find the maximum easily and then the second way is use former I'm not in the first way because it is not easy to compute. Thiss you know these contents So I'm going to use the formula. Dan, what's a foreigner? So when you have a function about thiss form, then you can easily and find the maximum. It is scary out ofthe spirit of thirteen. How did I do that? It is. It's karaoke east number, close piece number, and then take us in spirit. Right? So in general, if you have a constant in me, then you're mixed Mom will be scare it'LL ace care plus be scared, right? Yeah. Bye. And then by using their formal er, we can compute the maximum very easily. So scary toe one over G because I cured you. Yes, going to choose So it is point it away for right It is approximately By using calculator you can come into this So far so good, right? It's not that hard. And then that's when we want see first appearance off the pendulum So there is the time to complete back and one bank and forces tree again we can use for murder In this function we can easily find the period it iss to pry Oberst Cherie, How can I find this? So in general, if you have ah many times X in the trigger Elementary in functions break Then you are a period or B two Pi Siraj bhai, The absolute value Omi So in this case, our only that is scared of she. So we have thiss period, right? Nothing difficult So let's move on so d when will the pendulum first to be vertical? So if you look at this picture, this line is for ical So the problem asked you to find the time when this object pass through this point. So we're going to find the time. Oh, yeah. And at this point, he angle becomes you. So we have to find the time and she satisfies desiccation, set out key questions. Zero right? Then we can send this occasion Great. I just put Tio I just used to Fuller for his send off answerable t And then, uh since I want to pushed key collect tea in Oran term I divided by we'LL sign off Scared of g times t to collect tea in one term. Then we get this second occasion right here. What is this? Sign over a co sign. This is Hon. Gentle spirit of g times T right. Great. Then get in. And then here We have to think at this time a tasty she should be. How's chip? But if you find the solution on this one this one Look at this white course tenant's ex. If you remember by using Arc tangent, you confined Ah, point this point, right, well for him. But this is your never in point two times spirituality. Then you can easily find this point, however, the point is never Never Then we need a party in number so they have to line this point. So the difference between these two points is it's actually pie. So by heading high, we can get the exact solution two we'LL hear the sun becomes this so security g times ta Eucharist Our kind gentle negative points to times square to G class pie And by using calculator we can compute titties approximately. Then you get to point five pate. Am I right? Yes, this's right. Now this is an approximation. Approximately solution. So approximately tour Ah, you're going to five eight times later. But pendulum will be first. Well, first be vertical right now Let's move on. E, what's the angle of those to you? And the pendulum is for Cole. So at this point, we have to find the velocity, the angle realistic. Great. So to find the angle velocity, we have to differentiate Sarah and then pull out. Ah, just the exact time because you're kind. Yeah. So to plug this number Dan here, scourge of G times on scare I mean the tingle. I mean the tea we already find it isn't founded his soul, so to point five. So by plugging this number, we can compute it easily, then by using calculator peace ponies. Good point eight three. I mean, eight four. Right. So Ah, we can find an angle of others. See here. Notice that it is. The absolute value of this one is smaller than one, which is the initial angular velocity. Why is just happening? One? Because this is an approximately approximately solution. I mean, when you if you remember in the physics physics classes Teo, the potential energy applause, physics toe, Canary energy should be equal. So when your palate and pendulum will be the lowest, it should be fest. It shouldn't be fastest. Uh, since you used Teo approximate and more than not exact model. We have this research well, so thank you.