(a) What can you say about the graph of a solution of the...

(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 $

(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 $

Transcript

00:01 Okay, then i'm going to show you the differential equation.

00:04 What's the differential equation and the graph of the function and the solution of the differential equation? yeah, for the first one, as i show here, that one, is the differential equation.

00:27 It's not a derivative.

00:32 So normally we write d, y, dx.

00:37 So we write this one to this form.

00:49 So that one is not the, so as we in the calculus 1, we knew for the targeted derivative, here is x is the only variable on the right.

01:03 Here we have x and y.

01:05 So a little bit different.

01:08 So, but how can we, for problem one, what's the probability of this differential equation? we have no idea, so we need to solve the differential equation.

01:23 So if we write something here, and we take some, we put y in left side, x in right -hand side, and we take the integral on both sides, and we get this.

01:41 Then, okay, i suppose you see here.

01:49 Yeah, then we doing the simplification, we get this.

02:03 Then, so the whole idea is like we, because we don't want to, if we want to calculate the y -pine, equal to only one variable here, only if x, only one x, only one x, only the variable is x, we knew how to solve this problem.

02:29 It's similar with the calculates 1.

02:32 But here, okay, since we knew y, y, y, square rule, a square of y square is equal to this.

02:47 Then we go back to here, we knew y p.

02:51 Is x times y -cube.

02:53 So, we just need to solve the y.

02:56 It's this one.

03:00 And y is y is is this one...