Question
Consider the following separable initial value problem: 3=5 . 2 y (1) = -2 Determine the solution:v (T)
Consider the following separable initial value problem: 3=5 . 2 y (1) = -2 Determine the solution: v (T)
Answers
Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. $$\frac{d y}{d t}=t y+2, y(1)=2$$
Hey, so are your harsh allocation. Uh, tee times y penned equals one. So we'll rewrite Why? Prime of psd y over eat e So we hav e b y overheating Because what? Okay. And will do horrible separation. So we multiply both sides by DT viable Chase Beatty So have the way because one over t beauty. Okay, then get the anti girls boots sites. So the people of the wise Why the girl off all over t v T s l n off the absolute value 50. And then the arbitrary constancy Hey, were also given with initial condition. Why off one close to that means the value of why I went to one s to a so substituting these values are to get the value of C. We have to because Ln the absolute value of one plus a absolute value point, it's just zero there for season two. There go our solution. Yes. Why? Because l n off the absolute value off less. Okay,
All right. So our girlfriends elevation ISS Why prime a 50 Because still rely. So it's a bit simple. Different, Right. So what we're going to do is be right. Why? Prime of t s y overheats equals t over. Why? And then well, just separate ever variables. So multiply widen t on both sites we have Why do line? Because to meet you. Then get the anti derivatives boots sites we have Big Earl of why d why, that's why squared over two the girl off DVT We have to square over two and then don't forget the arbitrary constants c All right. And then after wards Hey afterwards moved get the equation terms of for Why so multiply both sides by two we have Why square multiply to here thing? So we have two times t squared over to less he and then get the square root of both sides We have why people squared off two times t squared over two Listen Hey and were given were the initial values that IHS Why? If one wants to So the value of why When 20162 So? So she should move this so ever General solutions. Solving for seat We have to call squared off to times one squared over two. Okay, so getting the square of boots sides, they have four plus two times. 1/2 Listen. Hey, and divide both sides by two. And so this one out you have to do because when I have and solving foresee. Hey, you subtract 1/2 on both sides of our creation Will have. See balls, see house or city over here. Therefore, our, uh our solution is why all squared off to times t square over to thus the Overton simplifying this one. We have a boy all squared off. Do times t squared plus three. Overton, notice we can cancel out to here. Finally, you have. Why both squared all t square less than Hey, So this is our final answer.
Were given this equation t times y prime of tea, which I'm gonna write his d Y over DT is equal to one. And then it tells us that why of one is equal to two and t is greater than zero. And it asked me to solve for y So what I'm gonna do, I'm gonna first apologised, separate my variables. So I'm going to get de y is equal to one over tee times DT that I'm gonna integrate both of those. So if I integrate both, I get why is equal to lawn t plus C? And then I have this initial value. So what I'm gonna do is I'm gonna say, Why is to when I have the lawn of one plus C right now one of 10 So what I get is two is equal to see. So that means my final answer is not just long of t plus C, but actually lot of t plus two. So that initial value allows me to actually find the constant
This question as to solve this D Y already t is equal to tee times y plus two. And the issue is that this is also not separable. Just like the previous question. If you watch that solution and the reason is not separable is because of this too. And so the tee times why are there? But I can't if I bring from the T. If I divide this by T or if I multiply it by t whatever you want to do to get T and d t by themselves, you can't. This, too, was always gonna have that extra variable in it. And so because it's t y plus two that plunged to is causing it to not be separable, which means we can't solve it using the methods we've been using.