Results 1 to 3 of 3

Thread: ODE-Lipschitz&Picard–Lindelöf theorem

  1. #1
    Member
    Joined
    Dec 2009
    Posts
    171

    ODE-Lipschitz&Picard–Lindelöf theorem

    Given This ODE:

    y' = (y-2) (x^2+y)^5
    y(0)=5

    A. Show that this problem has one solution that is defined in an open segment that contains 0.

    B. Let y(x) be a solution for this problem. Prove that y(x)>2 for every x in I and conclude that y'(x)>0 in I.
    Hint: You can use the solution of the problem: y'=(y-2)(x^2+y)^5 , y(x0)=2


    Help is needed !

    TNX!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Rebesques's Avatar
    Joined
    Jul 2005
    From
    My house.
    Posts
    658
    Thanks
    42
    For part a, as you mentioned,
    we can use the...

    Picard–Lindelöf theorem

    ...which says that, as $\displaystyle f(x,y)=(y-2)(x^2+y)^5$ is continuous in x and Lipschitz continuous in y, there is a unique solution to the initial value problem, which is defined in a neighbourhood $\displaystyle J$ of 0.

    Now for part b, note that $\displaystyle y_0=2$ is a solution of the equation. If $\displaystyle y$ was to meet $\displaystyle y_0$ somewhere over $\displaystyle J$, the uniqueness of the initial value problem $\displaystyle \{z'=f(x,z), \ z(0)=2, x\in \mathbb{R}\}$ would be violated. Since $\displaystyle y(0)>2$, we must have $\displaystyle y>2$ throughout $\displaystyle J$. Now the ode gives us a constant sign for $\displaystyle y'$ over $\displaystyle J$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2009
    Posts
    171
    Thanks a lot
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Picard Iteration
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: May 12th 2011, 07:51 AM
  2. picard's method of iteration
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Feb 14th 2010, 08:03 AM
  3. Picard's Iteration
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: Nov 23rd 2009, 01:41 PM
  4. Picard iter
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: Nov 10th 2009, 11:49 AM
  5. Picard's Iteration
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Feb 14th 2009, 09:42 AM

Search Tags


/mathhelpforum @mathhelpforum