Question

Could someone give me some tips for this kind of problem?
Determine a region of the xy-plane such that the given DE has an unique solution whose graph go through the point (x0, y0) in this region.

Answer 1

\[ \frac{dy}{dx} = y^{2/3} \]or \[ \frac{dy}{dx} = \sqrt{xy} \]

Answer 2

i'm sure it's DG-Universe lol

Answer 3

I know this shouldn't be exactly hard, but I don't know how to do it :-(

Answer 4

*bookmark

Answer 5

I don't have the answer for the second one. For the first one, it should be: semi-plan defined by y > 0 or y < 0.

Answer 6

Maybe solving the DE and checking where the solution is defined? In the existence and uniqueness theorem, we consider a rectangular region. So, I will try that.

Answer 7

it looks like xo,yo satisfies an initial condition right?

Answer 8

I thought so too, but all we are given about x0 y0 is that. I guess we have to solve for x0?

Answer 9

are these diffyQs on the same xy plane? or two different questions altogether?

Answer 10

two different questions.

Answer 11

I want help in any of these, so I can understand how to solve these kind of problem.

Answer 12

this* kind

Answer 13

then id assume that the solution is an interval on which the initial conditions are satisfied

theres no direction to make it a largest interval so any suitable one should suffice

Answer 14

the eqs seem rather separable to get a solution from

Answer 15

Hmm. You mean, for the first one my solution would be:
x + c = 3y^(1/3).
Then, I should pick > 0 or < 0 for this to hold?

Answer 16

I mean, if we pick x = 0, y = c, but I don't get why we can only pick y > 0 or y < 0

Answer 17

uniqueness implies that in the interval there is only one option that can result. right? without knowing more about the problem, that is my best idea

Answer 18

That's still confusing for me. And Boyce did not make this any simpler. Should I always disconsider the zero of the solution? Say, \( (4 - y^2)y ' = x^2 \). We have to pick y > 2 or y < -2 or -2 < y < 2.

Answer 19

Because if it's zero, we have infinite solutions?

Answer 20

the xo and yo conditions help to narrow down an interval, but so far i havent been able to interpret your conclusions.

4y - y^3/3 = x^3/3 + C , using xo and yo to calculate for C we can define a suitable y for this.

as such, I see no restrictions that would chop up the function into different intervals: no vertical asymptotes, no holes, no undefineds ....

Answer 21

Still, the answer provided by the book is the one I said above. :-( That's why it's so confusing for me.

Answer 22

maybe the explicit form has some issues tho

Answer 23

http://www.wolframalpha.com/input/?i=4y+-+y%5E3%2F3+%3D+x%5E3%2F3

Answer 24

notice that their y intervals chop up the graph into "functions" on the intervals

Answer 25

a solution has to be a function as well

Answer 26

So, I have to "graph" the solution and see where it's a function and if it's continuous (and if its y derivative is also continuous?)