Results 1 to 3 of 3

Math Help - Applying Cauchys theorem

  1. #1
    Newbie
    Joined
    Dec 2009
    Posts
    7

    Applying Cauchys theorem

    apply the existence and uniqueness theorem of Cauchy to the equation

    dy/dx = 1/y

    with the boundary condition

    y(0) = y

    to decide for what values of y the solution may or may not exist and be unique.

    im confused!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    A generic first order DE of the form...

    y^{'} = f(x,y) (1)

    ... with 'initial condition' y(x_{0})= y_{0} does admit one and only one solution if the so called 'Cauchy-Lipschitz conditions' are satisfied. In particular f(*,*) must be continuous and with bounded partial derivative respect to y in a 'small region' around [x_{0},y_{0}]. In this case is f(x,y) = \frac{1}{y} and that means that any 'initial condition' with y_{0}\ne 0 satisfies the conditions to have only and only one solution...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2009
    From
    Valparaíso, Chile
    Posts
    9
    In fact, you can relax the continuity of the partial derivative respecto to y (it isn't necessary that f be differentiable). If f satisfy a local Lipschitz condition in the variable y, the statement of the theorem also holds.
    Cya
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Applying Mean Value Theorem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 16th 2011, 03:10 AM
  2. Applying Stokes theorem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 31st 2009, 08:41 AM
  3. Applying the pythagoren theorem
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: April 24th 2009, 02:58 PM
  4. Applying De Moivres theorem..
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: December 4th 2008, 06:05 AM
  5. Cauchys Theorem
    Posted in the Calculus Forum
    Replies: 5
    Last Post: June 15th 2007, 06:39 AM

Search Tags


/mathhelpforum @mathhelpforum