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Let $k$, $l$ be smooth functions from an interval $I$ into $\mathbb R$ and $k>0$. Let's consider system of differential equations

$$ t'=k n, $$ $$ n'=-k t-l b, $$ $$ b'=ln $$ with unknow functions $t,n,b: I\rightarrow \mathbb R^3$.

How to show that scalar products below are zero: $$ t\cdot t'=0, $$ $$ n\cdot n'=0, $$ $$ b\cdot b'=0. $$

Thanks

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    I don't think you can. But from the given context, I am guessing that those conditions are assumed to hold initially, and you wish to show that they continue to hold. Am I right?2012-11-30
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    I wish to show that Euclidean norms of functions $t,n,b$ are constant. Then if we assume that for fixed $s_0$ norms of $t(s_0),n(s_0),b(s_0)$ is $1$, we obtain that this norms are equal $1$ at each point.2012-11-30

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