Show that y'=3x^3 -2xy has a unique solution through each point (a,b) and that y=e^-x^2+x^2-1 is the family of all solutions.

- UsukiDoll

- Stacey Warren - Expert brainly.com

Hey! We 've verified this expert answer for you, click below to unlock the details :)

- jamiebookeater

I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!

- UsukiDoll

|dw:1378583688707:dw|

- UsukiDoll

so the first thing that I want to do is to use separate variables for y'=3x^3-2xy because I was solving some problems that were identical to this and I had to use separate variables and solve for C by the given point

- experimentX

i think there is a theorem for that ... uniqueness and existence theorem.

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- UsukiDoll

yes that's the one.

- UsukiDoll

I have to prove it. I can solve something identical to it..it's just that the proof part is driving me nuts

- experimentX

use the condition from that theorem

- UsukiDoll

the theorem states that if F and Fy are continuous at some square S centered at (a,b) then there is a function y defined on some interval I around a such that y(1)=b and y'(x) = F (x,g(x)) for all X in I. Moreover, if g is any function such that g(a) = b and g'(x) = F(x,g(x))for X in I then g(x)=y(x) for x in I

- UsukiDoll

so I guess from point a to point b there is a solution somewhere in that spot.

- UsukiDoll

like....|dw:1378584100395:dw|

- experimentX

can you give me link of this theorem?

- UsukiDoll

can I give you the ebook link? It's on page 13 when you download it

- experimentX

google it ... most likely you will find on wiki

- UsukiDoll

http://www.math.uiuc.edu/~tyson/existence.pdf

- UsukiDoll

wait a sec....maybe y' = 3x^3-2xy is continuous because there is no restriction. It's not like there's a denominator at the bottom in which case it can't be 0

- experimentX

of course where is discontinuity?

- UsukiDoll

no where. at least not in the function y' =3x^3-2xy.
and not for ce^-x^2+X^2-1

- UsukiDoll

hmmm graph the function at point a b?!

- UsukiDoll

hold up. y=e^-x^2+x^2-1 must be related to y'=3x^3-2xy

- experimentX

y's is continuous of whole \( \Bbb R \times \Bbb R\)

- UsukiDoll

aww man missed the C

- experimentX

http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem

- UsukiDoll

|dw:1378584795067:dw|
for (a,b)
b = ce^-a^2+a^2-1

- UsukiDoll

|dw:1378584841499:dw|

- UsukiDoll

???????????????????

- experimentX

Ugh ... analysis hurt my eyes :(((

- UsukiDoll

How do I apply the theorem to this? like for any point (a,b) there is a number c such that b=ce-a^2+a^2-1?

- UsukiDoll

getting the partial derivative for Fy I get -2x

- experimentX

hold on a sec ... type 1 Dirac

- UsukiDoll

eh?

- UsukiDoll

1 Dirac ?

- experimentX

1 dirac = 1 word per hour

- UsukiDoll

oh I'm going too fast. sorry

- experimentX

http://en.wikipedia.org/wiki/Paul_Dirac#Personality

- UsukiDoll

the pdf http://www.math.uiuc.edu/~tyson/existence.pdf is telling me to do antiderivative.... so a solution must exist at point (a,b)

- UsukiDoll

separate the variables and then antiderivative. ..

- UsukiDoll

so..y'=3x^3-2xy at y(a)=b
Fy = -2x

- UsukiDoll

separate the variables. . ..

- experimentX

to show uniqueness, you just need to show that F(x,y) and F_y(x,y) is continuous on the point.

- UsukiDoll

continuous on the point (a,b)

- experimentX

on all domain

- UsukiDoll

|dw:1378585878614:dw|

- UsukiDoll

|dw:1378585915849:dw|

- UsukiDoll

this is where I may have screwed up a bit....maybe i should just +y on both sides to get y to the dy

- UsukiDoll

|dw:1378586014065:dw|

- UsukiDoll

or...|dw:1378586072354:dw|

- experimentX

\[ y' = 3x^3 -2xy = F(x,y) \]
how do you know \( F(x,y) \) is continuous everywhere?

- UsukiDoll

there's no denominator. that's one 2.... Fy = -2x because I have to take the partial derivative with respect to y

- UsukiDoll

graph the equations???

- experimentX

if you haven't gone through analysis on multivariable calculus then that logic is okay. but it's quite crazier on multivariable calculus than on single variable.

- UsukiDoll

I don't think I did. . .
hmmm the two equations are continuous at point (a,b) but how to show that? graph it out?

- experimentX

you need to show that the double limit is equal to it's function value
\[ \lim_{(x,y)\to (a,b)} F(x,y) = F(a,b) \]

- UsukiDoll

can we work on this together? like step by step on how this works?

- experimentX

F(x,y) = something = surface, not curve like in single variable calculus.

- UsukiDoll

I'm new at differential equations proofs....hence this is insane

- UsukiDoll

F(x,y) = 3x^3-2xy=Ce^x-2+x^2-1

- experimentX

no ... this ain't differential equations.
you should show that this surface is continuous everywhere.
http://www.wolframalpha.com/input/?i=plot+F%28x%2Cy%29%3D3x%5E3+-2xy

- UsukiDoll

rfor some reason that problem is in my differential equations book

- experimentX

in single variable you show that
lim x->x_0 f(x) = f(x_0) --> means continuity
you should show same here.

- UsukiDoll

show continuity on y'=3x^3-2xy

- experimentX

3x^3-2xy <-- only this one

- UsukiDoll

oh! so show continuity on 3x^3-2xy at (a,b)?

- experimentX

yes

- UsukiDoll

right after I eat something...pancakes are calling me or a pizza. . .

- experimentX

anytime you like ... but I might not be online for long.

- UsukiDoll

oh crud. . . so what do i do after I show continuity of that function?

- experimentX

(by existence theorem) it proves that the solution exists.
then F(x,y) = =3x^3 -2xy
F_y = -2x <-- show it is continuous.
By uniqueness theorem, the solution is unique.

- UsukiDoll

-.- I had that Fy = -2x earlier. oh awlz I went in circles.

- experimentX

because you did not pay attention

- UsukiDoll

so what does that y = ce^-x^2+x^2-1 stuff have to do with the first function?

- experimentX

yes ... show that it is the solution of DE

- UsukiDoll

plug it in the original equation?

- experimentX

yes that would do that too.

- UsukiDoll

so that would be y'=3x^3-2x(ce^-x^2+x^2-1)

- experimentX

Eh ... where is the differentiate site?

- experimentX

*side

- experimentX

3x^3-2x(ce^-x^2+x^2-1) = d/dx (ce^-x^2+x^2-1)

- UsukiDoll

antiderivative of y' and use separate variables.

- experimentX

hmm ...

- experimentX

just simplify this ... and show that equality holds.
3x^3-2x(ce^-x^2+x^2-1) = d/dx (ce^-x^2+x^2-1)

- experimentX

http://www.wolframalpha.com/input/?i=+y%27%3D3x%5E3+-2xy+

- UsukiDoll

simplifying as like combining like terms

- experimentX

yes yes ... but equality will not hold since you put it wrong on your question.

- experimentX

or just solve it, y'=3x^3 -2xy .. can you?

- UsukiDoll

yeah I forgot the C on the original question part.

- UsukiDoll

solve y'=3x^3-2xy??? by separate variables? antiderivative?

- experimentX

No ... this is first order linear equation.

- experimentX

yes to anti-derivative /// no to separate variables.

- UsukiDoll

do i put it in the form dy/dx+P(x) = q(x)

- experimentX

yes

- UsukiDoll

alright :). I dont' understand why my vintage ebook doesn't even stress that part since it's critical. thanks. Now I shall munchie.

- experimentX

ok ... I am going to sleep

- UsukiDoll

nighty night. It's daylight where I live. sweet dreams.

- experimentX

it's morning where I live ... i go to sleep in the morning.

- experimentX

have a nice day.

Looking for something else?

Not the answer you are looking for? Search for more explanations.