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Thread: Quiestion to do with existence and uniqueness of a equation

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

    Hi

    My question is
    Consider the initial value problem

    $\displaystyle yy'=x-1$

    subject to the intitial condition $\displaystyle y(x=X)=Y$ 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 $\displaystyle x$.

    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'...

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

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

    The DE can be written in the form...

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

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

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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