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Math Help - Quiestion to do with existence and uniqueness of a equation

  1. #1
    Junior Member
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    Quiestion to do with existence and uniqueness of a equation

    Hi

    My question is
    Consider the initial value problem

    yy'=x-1

    subject to the intitial condition y(x=X)=Y where X and Y are constants.

    a)For which values of X and Y does the existence and uniqueness theorem ensure the existence of a unique solution? If a unique solution can be shown to exist, will it exist for all values of x.

    b)Find the general solution of the ODE, using any suitable method.

    In my notes i have a section on uniqueness of solutions but its not very detailed. It mentions Peano existence theorem and also mentions picard existence theorem.

    I wanted to know what the difference what between these two theorems also.

    thanks
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  2. #2
    MHF Contributor chisigma's Avatar
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    If we write the DE in 'standard form'...

    y^{'}= f(x,y) , y(x_{0})=y_{0} (1)

    ... the solution exists and is unique if f(*,*) is 'Lipschitz continous' in (x_{0}, y_{0}) and clearly that is not verified for y_{0}=0...

    The DE can be written in the form...

    y\cdot dy = (x-1)\cdot dx (2)

    ... so that the variables are 'separated' and the DE can be solved with simple integration...

    Kind regards

    \chi \sigma
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