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Linear ODE with constant coefficients

$$a_oy^{(n)}+…+a_ny=0$$

has a nontrivial constant solution. Find $a_n$.

I've never done anything like this before and I have no clue how to do this. How do I solve this problem? Thanks.

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    $\begingroup$ What have you tried? Have you tried solving it for $n = 1, 2, 3...$? Do you see a pattern? It really helps to share your thoughts and what you have done to get an idea of where you are at. Regards $\endgroup$ – Amzoti Oct 2 '13 at 2:19
  • $\begingroup$ I know $a_n$, $a_{n+1}$, etc. are supposed to be constant. Does that mean $a_n$ has infinite solutions? $\endgroup$ – user91971 Oct 2 '13 at 12:28
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A nontrivial constant solution means that there is a function $y$ which

  1. satisfies the equation
  2. is constant
  3. that constant is not zero

All derivatives $y'$, $y''$, etc of a constant function are equal to $0$. Therefore, the equation takes the form $$a_0 \cdot 0+a_1 \cdot 0+ \cdots + a_{n-1} \cdot 0+ a_n y = 0$$ Simplify this. Recall that $y$ is not zero. Make your conclusion about $a_n$.

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  • $\begingroup$ Ah, I see. Thanks! $\endgroup$ – user91971 Oct 3 '13 at 1:16

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