Differential Equations

An ordinary equation like $x^2 = 4$ asks for a number. A differential equation asks for a function, and it pins that function down through its derivatives. The simplest one in finance is continuous compounding. A balance $M(t)$ earning rate $r$ grows at a speed proportional to its size:

The order of the equation is the highest derivative in it, so this one is first order. Read it as a rule for the slope: wherever the balance is, the equation says how fast it is moving. Draw a short segment with that slope at every point of the $(t, M)$ plane and you get a slope field. A solution is any curve that follows the segments.

Many curves do. $M(t) = Ce^{rt}$ works for every constant $C$, which you can check by differentiating: $(Ce^{rt})' = rCe^{rt}$. An initial condition such as $M(0) = 100$ picks out one of them, here $C = 100$. At $r = 5\%$ the balance after 10 years is $100e^{0.5} \approx 164.87$, and it doubles when $e^{0.05t} = 2$, at $t = \ln 2/0.05 \approx 13.9$ years. This is the rule of 70: since $100\ln 2 \approx 69.3$, the doubling time in years is about 70 divided by the rate in percent.

Checking a proposed solution is always easy. Substitute it and see whether both sides agree. If you blank on a method in an interview, guessing a form (an exponential $e^{st}$, a constant, a polynomial) and plugging it in often gets you the answer anyway.