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Problem 8 Easy Difficulty

(a) What can you say about the graph of a solution of the equation $ y^{'} = xy^3 $ when $ x $ is close to $ 0? $ What if $ x $ is large?

(b) Verify that all members of the family $ y = (c - x^2) ^{{-}{1/2}} $ are solutions of the differential equation $ y^{'} = xy^3. $

(c) Graph several members of the family of solutions on a common screen. Do the graphs confirm what you predicted in part (a)?

(d) Find a solution of the initial-value problem

$ y^{'} = xy^3 $ $ y(0) = 2 $

Answer

(a) When $x$ is close to zero, slope of the function is close to zero.
When $x$ becomes as large, slope of the function also becomes large in magnitude.
(b) Solutions of the form $y=\left(c-x^{2}\right)^{-1 / 2}$ solve equations of the form $y^{\prime}=x y^{3}$
(c) See Graph
(d) $\frac{y^{2}\left(4-x^{2}\right)}{x^{2}}=-4$

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Video Transcript

Okay, then I'm going to show you the differential equation. What's the differential equation and the graph off the funk at the solution and and solution off the differential creation? Yeah. For the first one. Yes. Yes, I show here, that one. It's the differential equation. It's not a direct pretty. So normally we write d Y d x two. So we write this one to you. This this form. So if there one is not the so as we in the calculus one we knew for the target derivative here is X is the only variable on the right. Here we have X. And why so a little bit different. So But how can we? For problem one. What's the property off the off this differential equation? We have no idea. So we need to solve the differential equation. So if we write something here and we take some, we put white in left side vaccine right hand side, and we take the integral on both sides and we give these, then Okay, I suppose you see here. Yeah, Sam, with doing the simple simplification, we get this sand. So the whole idea is like we because we don't want to if we want to calculate the white point Echoed in equal to only one variable Here only you've X only one x here Only the variable is X. We knew how he solves this problem. A similar result calculates one are here. Okay, since we knew why Why? Why? Why? Square drew a square off why square is equal to these. Then we go back to here. We knew why? Point is X times y cube. So we just need to stop the why is this one and why east? So this one? Yeah, here is some calculation. Yeah, and we saw the white pine equal to thes Here is so ra wearable Here is X So when we take the limit, exit goes to stereo We get holders in zero, right? Just like c o d y by C minus CEO Is he right? San if x goes to large So when we back to hear this equation and when x x large. But how large is it if the so we need to limit the X So we knew this equation And for this equation Because why everything square should be right And this one cannot be zero since this one is in the denominator. So we get this and we need it here. Then we take the name it. So for the graph, we can sing Think Here. So here. If X goes to square off, Positive square off. See, We get face if we take If they go to the minor square off. See, we have the same thing. So it's like this Here is the wife y graph. Okay for the part B. How can we prove this one? Satisfy the The solution is satisfy the differential equation the easiest. We just need to parking. Yeah, it's here. Just take calculated the relative. Yeah, they are equal. Yeah. Satisfied? Here is the graph we'll see. Sees from minus minus, minus 10 to 10. You can use the mathematics to go. I used the on line one. It's kind of like the shooting message in physics. So for the shooting message, we we have one point and we use the computer program to run from this one to find the sea. Then here is from the sea. Just need to program one point. What? X equal to zero. Why equal to two. So we found C. He sees, and we plug in the solution. Thank you.