1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Problem for Theorem of Uniqueness

  1. Aug 15, 2011 #1
    1. The problem statement, all variables and given/known data

    Check if the given initial value problem has a unique solution

    2. Relevant equations

    y'=y^(1/2), y(4)=0

    3. The attempt at a solution

    f=y^(1/2) and its partial derivative 1/2(root of y) are continuous except where y<=0. We can take any rectangle R containing the initial value point (4,0). Then the hypothesis of theorem of uniqueness is satisfied.

    I want to make sure if this way is correct. Some help please.

    Thanks!
     
  2. jcsd
  3. Aug 15, 2011 #2

    hunt_mat

    User Avatar
    Homework Helper

    The usual way you tackle this kind of thing is proof by contradiction and examine the integral:
    [tex]
    \int \left| \frac{dw}{dx}\right|^{2}dx
    [/tex]
    where [itex]w=u-v[/itex] and u and v solve the original equation.

    I think in this case though, you can solve the general equation (without using the iniitial/boundary condition) and show that it's solution depends upon a single parameter. That parameter is determined with the condition you were given.
     
  4. Aug 15, 2011 #3
    Did you mean that my way and answer is wrong? This problem should be verified by using the theorem of uniqueness.

    I got y(t)=(4x^2)-64. Then can i say that this is a unique solution?
     
  5. Aug 15, 2011 #4

    hunt_mat

    User Avatar
    Homework Helper

    As I don't know the theorem you're using I can't say either way.
     
  6. Aug 15, 2011 #5
    sorry, you're right. The theorem I was trying to use is that if f and its partial derivative is continuous on the rectangle containing the given initial value problem, we can say that it has a unique theorem.
     
  7. Aug 15, 2011 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    It would help if you actually stated the theorem you are using. I suspect you are using
    "The initial value problem y'= f(x,y), [itex]y(x_0)= y_0[/itex], has a unique solution in some neighborhood of the point [itex](x_0, y_0)[/itex] if both f(x,y) and [itex]f_y(x,y)[/itex] are continuous in some neighborhood of that point."
    No, you cannot "take any rectangle R containing the initial value point (4, 0)" because that will necessarily include points where y<= 0, where the functions are NOT continuous.

     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Problem for Theorem of Uniqueness
Loading...