I understand that $\frac{dy}{dx} = k*y$ and when $k>0$ this is the law of natural growth and when $k<0%$ this is the law of natural decay, but my textbook gives an example of radioactive decay as follows which confuses me:
Radioactive substances decay by spontaneously emitting radiation. If is the mass remaining from an initial mass of the substance after time t, then the relative decay rate
$\frac{-1}{m}\frac{dm}{dt}$ (1)
has been found experimentally to be constant. (Since $\frac{dm}{dt}$ is negative, the relative decay rate is positive.) It follows that:
$\frac{dm}{dt}=km$ (2)
where k is a negative constant. In other words, radioactive substances decay at a rate proportional to the remaining mass. This means that we can use to show that the mass
decays exponentially:
$m(t)=m(0)e^{kt}$ (This equation I understand and accept, the above two confuses me)
Now eventually when the example becomes numerical, then of course k becomes a negative number since it is natural decay. However, the above general notation explanation confuses me because it keeps changing the sign of $k$=relative decay rate(from negative(1) to positive(2), but eventually when numerically worked it turns out to be a negative constant since it's natural decay). I know that $k$ must be less than zero for decay but I'm just trying to fully grasp the notation signs that the textbook uses in the explanation above. Please clarify this. Thank you.
Best Answer
On one hand you can say k is defined positive. Then -k is negative. The differential equation in case of decay is then $\frac{dm}{dt}=-km$. The solution then is $m=C\cdot e^{-kt}$
On the other hand you can say that k can be positive or negative. Then the differential eqution is $\frac{dm}{dt}=km$ The solution then is $m=C\cdot e^{kt}$ If you have an exponential decay k gets negative.
In both cases you get a negative expoenent, if you have an exponential decay.