# systems of linear differential equations

Solve:

$y_1'=12y_3$

$y_2'=y_1+13y3$

$y_3'=y_2$

so the solution is:

$y_1=3c_1e^{4x}-4c_2e^{-3x}-12c_3e^{-x}$

$y_1=4c_1e^{4x}-3c_2e^{-3x}-c_3e^{-x}$

$y_1=1c_1e^{4x}+c_2e^{-3x}+c_3e^{-x}$

now my question is: find a value y$(0)$ other than y$(0)$$=(0,0,0)$ such that y$(x)$ approaches $(0,0,0)$as $x$ approaches infinity.

I'm not even sure what that means ><

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It should be obvious that the first term in each expression will "blow up" as $x\rightarrow\infty$. So you must choose $c_1=0$.
For the second and third terms, these both approach $0$ as $x\rightarrow\infty$, so any choice of $c_2$ and $c_3$ will satisfy the condition which you state.