Existence and Uniqueness Theorem for Linear Equations

**alexmahone** · September 30th 2011, 11:41 AM

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 $y_1,\ y_2,\ ...,\ y_n$ .

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

$y_i^{(i-1)}(a)=1$ and $y_i^{(k)}(a)=0$ if $k\neq i-1$ .)

**alexmahone** · October 1st 2011, 01:47 AM

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

Evaluating at $a$ ,

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

$c_1=0$

Differentiating once,

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

Evaluating at $a$ ,

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

$c_2=0$

etc.

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

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

Is this right?

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