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Thread: Existence and Uniqueness Theorem for Linear Equations

  1. #1
    MHF Contributor alexmahone's Avatar
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    Existence and Uniqueness Theorem for Linear Equations

    Prove that an nth-order homogenous linear differential equation satisfying the hypotheses of the Existence and Uniqueness Theorem for Linear Equations has n linearly independent solutions $\displaystyle y_1,\ y_2,\ ...,\ y_n$.

    (Suggestion: Let $\displaystyle y_i$ be the unique solution such that

    $\displaystyle y_i^{(i-1)}(a)=1$ and $\displaystyle y_i^{(k)}(a)=0$ if $\displaystyle k\neq i-1$.)
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Existence and Uniqueness Theorem for Linear Equations

    Assume $\displaystyle c_1y_1+c_2y_2+...+c_ny_n=0$.

    Evaluating at $\displaystyle a$,

    $\displaystyle c_1y_1(a)+c_2y_2(a)+...+c_ny_n(a)=0$

    $\displaystyle c_1=0$

    Differentiating once,

    $\displaystyle c_1y_1'+c_2y_2'+...+c_ny_n'=0$

    Evaluating at $\displaystyle a$,

    $\displaystyle c_1y_1'(a)+c_2y_2'(a)+...+c_ny_n'(a)=0$

    $\displaystyle c_2=0$

    etc.

    Thus, $\displaystyle c_1=c_2=c_3=...=c_n=0$.

    So, $\displaystyle y_1,\ y_2,\ ...,\ y_n$ are linearly independent.

    Is this right?
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