How do we derive the formula for continuous compounding by writing an ODE?

Question

Recently, I encountered a question on continuously compounding interest. The solution started with the following differential equation

$$\frac{dN}{dt} - rN = 0$$

where $r$ is the interest rate and $N$ is the principal amount. However, I couldn't understand how this equation was formulated. There was no explanation given. I know the solution to it is $

$$N (t) = N_{0} e^{rt}$$

which this equation correctly satisfies, but I couldn't find the proof / thinking online. All the proofs online involve taking the limit

$$N = N_{0}\lim_{n \to \infty} \left(1+\frac{r}{n}\right)^{tn}$$

Can someone please explain how exactly one arrives at the differential equation $\frac{dN}{dt}-rN=0$?

Answer 1

There is a simple calculus explanation:

It is based on the consideration that the percentage change of the principal $dN/N$ during an infinitesimal time period $dt$ is $r_t dt$, i.e.

$$\frac{dN}{N} = r_t dt\tag{1}$$

where the rate of change $r_t$ is the so-called instantaneous interest rate. The expression implies that The principal appreciation over each successive period $dt$ is built upon its accumulation over all previous periods, hence continuous compounding.

Rearrange (1) allows one to arrive at the deferential equation in question, i.e.

$$\frac{dN}{dt}-r_tN=0$$

Answer 2

The equation can be rearranged as $$\frac{\mathrm d N}{\mathrm d t}=rN,$$ which says that the rate of growth of the amount at an instant of time $t$ is proportional to the amount present.