Differential Equations

[Han]

Any calculus book will be a perfectly fine introduction into differential equations. Once you learn the concept of a (mathematical) derivative, then you have essentially all the tools to understand what a differential equation is. (Solving them, however, requires a bit more mathematics.)

Perhaps a finance example might be helpful. Consider an account that accrues an effective annual rate of 5%, and that any interest earned is reinvested. Suppose \( P \) represents the (future) value of the account after t years of investment. In mathematical terms, \( P \) is a function of \( t \); the notation \( P(t) \) is generally used. For brevity, mathematicians may simply write \( P \) when it is understood that \( t \) is the independent variable. The average annual rate of change in \( P \), which is essentially the interest earned over a one-year period, is
\[ \underbrace{\frac{\Delta P}{\Delta t}}_{\text{interest}} = (\text{interest rate}) \times (\text{current value of account}) =
0.05 \cdot P \]
The left hand side is an approximation of what is mathematically called the "derivative" of the function \( P \). The derivative of a function measures "instantaneous" rates of change as opposed to an "average rate of change." An easy analogy would be driving at an average speed of 60mph over a 1 hour period, whereas at any given moment one's actual (instantaneous) speed may be slightly above or below 60mph (i.e. not necessarily 60mph, but relatively close to it "most of the time").

If we instead looked at smaller growth periods (perhaps interest compounded not annually, but monthly, or weekly, or daily, or hourly, etc.) then the average rate of change more closely resembles the instantaneous rate of change (the derivative). In the interest example above, this would be equivalent to looking at interest compounded continuously (with a nominal rate that is slightly less than 5%).

The common notation for a derivative (using the example above) is \( \displaystyle \frac{dP}{dt} \), or \( P'(t) \), or \( P' \) for brevity.

Anyway, a differential equation is merely an equation that involves a derivative of any sort. Solving a differential equation amounts to finding a function whose derivative satisfies the differential equation. To use another (finance) example, consider the formula for simple interest. Let \( P_0 \) denote the present value, and \( P(t) \) be the future value with an annual rate of \( r \) over \( t \) years. The formula is
\[ P(t) = P_0 + P_0 \cdot r \cdot t \]
and \( P(t) \) satisfies the differential equation
\[ \frac{dP}{dt} = P_0 \cdot r \]
with an initial condition of \( P(0) = P_0 + P_0 \cdot r \cdot 0 = P_0 \) (i.e. the present value is \( P_0 \)).