UsukiDoll
  • UsukiDoll
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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UsukiDoll
  • UsukiDoll
|dw:1378583688707:dw|
UsukiDoll
  • 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
  • experimentX
i think there is a theorem for that ... uniqueness and existence theorem.

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More answers

UsukiDoll
  • UsukiDoll
yes that's the one.
UsukiDoll
  • 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
  • experimentX
use the condition from that theorem
UsukiDoll
  • 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
  • UsukiDoll
so I guess from point a to point b there is a solution somewhere in that spot.
UsukiDoll
  • UsukiDoll
like....|dw:1378584100395:dw|
experimentX
  • experimentX
can you give me link of this theorem?
UsukiDoll
  • UsukiDoll
can I give you the ebook link? It's on page 13 when you download it
experimentX
  • experimentX
google it ... most likely you will find on wiki
UsukiDoll
  • UsukiDoll
http://www.math.uiuc.edu/~tyson/existence.pdf
UsukiDoll
  • 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
  • experimentX
of course where is discontinuity?
UsukiDoll
  • UsukiDoll
no where. at least not in the function y' =3x^3-2xy. and not for ce^-x^2+X^2-1
UsukiDoll
  • UsukiDoll
hmmm graph the function at point a b?!
UsukiDoll
  • UsukiDoll
hold up. y=e^-x^2+x^2-1 must be related to y'=3x^3-2xy
experimentX
  • experimentX
y's is continuous of whole \( \Bbb R \times \Bbb R\)
UsukiDoll
  • UsukiDoll
aww man missed the C
experimentX
  • experimentX
http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
UsukiDoll
  • UsukiDoll
|dw:1378584795067:dw| for (a,b) b = ce^-a^2+a^2-1
UsukiDoll
  • UsukiDoll
|dw:1378584841499:dw|
UsukiDoll
  • UsukiDoll
???????????????????
experimentX
  • experimentX
Ugh ... analysis hurt my eyes :(((
UsukiDoll
  • 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
  • UsukiDoll
getting the partial derivative for Fy I get -2x
experimentX
  • experimentX
hold on a sec ... type 1 Dirac
UsukiDoll
  • UsukiDoll
eh?
UsukiDoll
  • UsukiDoll
1 Dirac ?
experimentX
  • experimentX
1 dirac = 1 word per hour
UsukiDoll
  • UsukiDoll
oh I'm going too fast. sorry
experimentX
  • experimentX
http://en.wikipedia.org/wiki/Paul_Dirac#Personality
UsukiDoll
  • 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
  • UsukiDoll
separate the variables and then antiderivative. ..
UsukiDoll
  • UsukiDoll
so..y'=3x^3-2xy at y(a)=b Fy = -2x
UsukiDoll
  • UsukiDoll
separate the variables. . ..
experimentX
  • experimentX
to show uniqueness, you just need to show that F(x,y) and F_y(x,y) is continuous on the point.
UsukiDoll
  • UsukiDoll
continuous on the point (a,b)
experimentX
  • experimentX
on all domain
UsukiDoll
  • UsukiDoll
|dw:1378585878614:dw|
UsukiDoll
  • UsukiDoll
|dw:1378585915849:dw|
UsukiDoll
  • 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
  • UsukiDoll
|dw:1378586014065:dw|
UsukiDoll
  • UsukiDoll
or...|dw:1378586072354:dw|
experimentX
  • experimentX
\[ y' = 3x^3 -2xy = F(x,y) \] how do you know \( F(x,y) \) is continuous everywhere?
UsukiDoll
  • UsukiDoll
there's no denominator. that's one 2.... Fy = -2x because I have to take the partial derivative with respect to y
UsukiDoll
  • UsukiDoll
graph the equations???
experimentX
  • 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
  • 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
  • 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
  • UsukiDoll
can we work on this together? like step by step on how this works?
experimentX
  • experimentX
F(x,y) = something = surface, not curve like in single variable calculus.
UsukiDoll
  • UsukiDoll
I'm new at differential equations proofs....hence this is insane
UsukiDoll
  • UsukiDoll
F(x,y) = 3x^3-2xy=Ce^x-2+x^2-1
experimentX
  • 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
  • UsukiDoll
rfor some reason that problem is in my differential equations book
experimentX
  • experimentX
in single variable you show that lim x->x_0 f(x) = f(x_0) --> means continuity you should show same here.
UsukiDoll
  • UsukiDoll
show continuity on y'=3x^3-2xy
experimentX
  • experimentX
3x^3-2xy <-- only this one
UsukiDoll
  • UsukiDoll
oh! so show continuity on 3x^3-2xy at (a,b)?
experimentX
  • experimentX
yes
UsukiDoll
  • UsukiDoll
right after I eat something...pancakes are calling me or a pizza. . .
experimentX
  • experimentX
anytime you like ... but I might not be online for long.
UsukiDoll
  • UsukiDoll
oh crud. . . so what do i do after I show continuity of that function?
experimentX
  • 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
  • UsukiDoll
-.- I had that Fy = -2x earlier. oh awlz I went in circles.
experimentX
  • experimentX
because you did not pay attention
UsukiDoll
  • UsukiDoll
so what does that y = ce^-x^2+x^2-1 stuff have to do with the first function?
experimentX
  • experimentX
yes ... show that it is the solution of DE
UsukiDoll
  • UsukiDoll
plug it in the original equation?
experimentX
  • experimentX
yes that would do that too.
UsukiDoll
  • UsukiDoll
so that would be y'=3x^3-2x(ce^-x^2+x^2-1)
experimentX
  • experimentX
Eh ... where is the differentiate site?
experimentX
  • experimentX
*side
experimentX
  • experimentX
3x^3-2x(ce^-x^2+x^2-1) = d/dx (ce^-x^2+x^2-1)
UsukiDoll
  • UsukiDoll
antiderivative of y' and use separate variables.
experimentX
  • experimentX
hmm ...
experimentX
  • 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
  • experimentX
http://www.wolframalpha.com/input/?i=+y%27%3D3x%5E3+-2xy+
UsukiDoll
  • UsukiDoll
simplifying as like combining like terms
experimentX
  • experimentX
yes yes ... but equality will not hold since you put it wrong on your question.
experimentX
  • experimentX
or just solve it, y'=3x^3 -2xy .. can you?
UsukiDoll
  • UsukiDoll
yeah I forgot the C on the original question part.
UsukiDoll
  • UsukiDoll
solve y'=3x^3-2xy??? by separate variables? antiderivative?
experimentX
  • experimentX
No ... this is first order linear equation.
experimentX
  • experimentX
yes to anti-derivative /// no to separate variables.
UsukiDoll
  • UsukiDoll
do i put it in the form dy/dx+P(x) = q(x)
experimentX
  • experimentX
yes
UsukiDoll
  • 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
  • experimentX
ok ... I am going to sleep
UsukiDoll
  • UsukiDoll
nighty night. It's daylight where I live. sweet dreams.
experimentX
  • experimentX
it's morning where I live ... i go to sleep in the morning.
experimentX
  • experimentX
have a nice day.

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