# Finding a formula from a mathematical model.

I don't understand how they've constructed the differential equation below. Can someone explain how the differential equation relates to the given constraints in the problem?

An explanation in words will be appreciated.

• Which step is giving you problems?
– 5xum
Oct 8 '18 at 12:47
• The rate of growth in quantity $P$ is typically defined as $$\frac{\frac{dP}{dt}}{P}$$ The differential equation follows from that.
– Paul
Oct 8 '18 at 12:51
• @5xum The first step where they actually fabricate the differential equation. Oct 8 '18 at 13:05
• Constant growth rate gives exponential time rise for $P$ Oct 8 '18 at 16:02

Everything is actually written out in the text. The growth rate is given in percentages. So, it's not "how many dollars per time unit" but "in what proportion to the previous value per time unit". In this sense, the differential equation is already given by the definition of given growth rate.

It's easiest to write out "small differences" and then recognize the limit to differentials. Imagine a small time interval $$\Delta t$$ in which the interest is applied. Percentage means relative change with respect to current value. So, the given information has to be interpreted this way:

$$\frac{\Delta P}{P}=(2+3\cos\frac{t}{2})\frac{1}{100}\Delta t$$

We just wrote out what "percentage rate" means. Now, rearrange and send $$\Delta \to d$$ and you are done.

p.s. It is much easier to integrate using a definite integral instead of taking extra steps to find the integrating constant. Just integrate from initial to final time on one side and from initial to final $$P$$ on the other:

The integral goes:

$$\int_{P_0}^P \frac{dP}{P}=\int_0^t (2+3\cos\frac{t}{2})\frac{1}{100}dt$$ and you have the result in a single step.