Recognising and applying differential equations

One of the important skills related to differential equations is being able to choose an appropriate equation to model a real-world system. Once we do this, we can then use this model to answer some complex questions about these systems. For example:

A scientist has extracted $18$ milligrams of an a radioactive isotope, which is decaying so that the mass $m(t)$ of the isotope remaining decreases over time. The rate at which the isotope decays is proportional to the mass of the isotope present.

We can write a differential equation to model the change in mass of the isotope over time. The rate of change of the isotope's mass can be written as $\frac{dm}{dt}$ .

We have been told the the rate of change is proportional to the mass present, which means that

$\frac{dm}{dt} = km$

We also know that the mass is decreasing over time, so $k < 0$ . We also know that initially, the scientist has $18$ grams of the isotope, so $m(0) = 18$ . Now that we have constructed the differential equation, we can answer some interesting questions about it.

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