Formation of Differential Equations from Physical Situations

Physical laws that state how a quantity's rate of change is related to other quantities translate immediately into a differential equation — no arbitrary constants need to be eliminated, because the law itself is already stated in terms of a rate.

Model 1 (Newton's Law). By Newton's second law, $F=ma$: force equals mass times the instantaneous acceleration. For an object of constant mass $m$ released in free fall from height $h(t)$ above the ground, this becomes the second-order differential equation

where $f$ packages whatever forces (gravity, air resistance, …) act on the object — the unknown function is the height $h$ as a function of time $t$.

Model 2 (Population Growth / Malthusian Law). A population grows because its members reproduce — more individuals present yield more offspring, so the population increases faster as it gets larger. If the rate of growth of a population's biomass $N(t)$ at time $t$ is directly proportional to the biomass already present, this gives the first-order equation

$\dfrac{dN}{dt}=rN,\qquad r>0\ \text{(the growth rate)}.$ …