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Based on a solution given as:

$x'' + 0.035x' + 0.00005x - 0.009 = 0$

Solve the characteristic equation. Based on your values of $r$, how large will $t$ have to be for the exponentials in the solution to have decayed to $2\%$ of their original value?

I solved the characteristic equation and got values of $r$ as $r = -0.0335$ and $r = -0.00149$. The solution then becomes:

$x(t) = c_1e^{-0.0335t}+c_2e^{-0.00149t}$

Now, how do I check what $t$ has to be for the exponentials to have decayed to $2\%$ of their value? Wouldn't I have to know $c_1$ and $c_2$?

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It is asking for the time when $\frac {x(t)}{x(0)}=0.02$ If you just had one of your terms, you wouldn't care about the starting value, as it would divide out. The term in $c_2$ is decaying more slowly than the term in $c_1$, so the worst case is when $c_1=0$ Then we want

$\exp(-0.00149t)=0.02 \\ -0.00149t=\ln 0.02 \\0.00149t=\ln 50 \\ t\approx 2625.5$