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Math Help - Existence and Uniqueness

  1. #1
    No one in Particular VonNemo19's Avatar
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    Existence and Uniqueness

    Hi everyone,

    I'm having trouble with the ideas involved with the uniquesness theorem for a linear nth order IVP.

    As an example:

    The initial value problem

    3y'''+5y''-y'+7y=0, y(1)=0, y'(1)=0, y''(1)=0.

    Now, I can see that y=0 is a trivial solution here, but by the theorem, this has got to be the ONLY solution on any interval containing 1. I don't see this. Is this true? Is there no other solution to this problem ?
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  2. #2
    MHF Contributor chisigma's Avatar
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    Let's suppose that the solution od the DE around x=1 is of the form...

    \displaystyle y(x)= \sum_{n=0}^{\infty} y^{(n)} (1)\ \frac{(x-1)^{n}}{n!} (1)

    ... and our scope is to find the y^{(n)}(1) for all n. The 'initial conditions' give us y(1)=y^{'}(1)=y^{''}(1)=0. The sucessive drivatives are...

    \displaystyle y^{(3)} (1)= - \frac{5}{3}\ y^{(2)} (1) +  \frac{1}{3}\ y^{(1)} (1) - \frac{7}{3}\ y(1)=0 (2)

    \displaystyle y^{(4)} (1) = - \frac{5}{3}\ y^{(3)} (1) + \frac{1}{3}\ y^{(2)} (1) - \frac{7}{3}\ y^{(1)} (1) =0 (3)

    ... and so one. All the derivatives in (1) vanish so that y=0 is the only solution...

    Kind regards

    \chi \sigma
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