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I have this system in maple and I can't get it work write:

sist:=diff(x(t), t) =-k*x(t), x(0)=x0;

Can someone help me get the desintegration constant k if the half-time is t1/2=700*10^6

Thanks

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The solution of the differential equation is $x(t)=x_0e^{-kt}.$

To say that the half-life is $h$ is to say that $x(h)=\frac{x_0}{2}$. Substituting in the formula for $x(t)$, we get $x_0e^{-kh}=\frac{x_0}{2}.$

Cancel the $x_0$, and take reciprocals (this part is not strictly necessary). We get $e^{kh}=2.$ Take the natural logarithm, which we will call $\ln$. We get $kh=\ln 2\quad\text{and therefore} \quad k=\frac{\ln 2}{h}.$ Now use the given value $700\times 10^6$ of the half-life. I get that $k$ is approximately $9.902\times 10^{-10}$.

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    @AndréNicolas Good to hear that. – 2016-09-04