What are differential equations useful for?

To provide a simple answer that does not involve a lot of mathematics, differential equations are used in almost every scientific field to provide a mathematical description of a process. They are useful because most processes can be described in terms of changes that result from small differences in elements of the process. When these small changes are represented in mathematical form the result is a differential equation. Because the changes are small, it is possible to make assumptions that simplify the expression of the process.

For example, consider the discharge of a capacitor into a resistor. We know that the charge stored in the capacitor is Q = V*C, where V=voltage on the capacitor, and C is the capacitance. A small change in charge, dQ will cause a small change in voltage, dV. The voltage across the resitor will be V, and the current in the resitor V/R. The current is the amount of charge divided by the time, or change in current is change in charge divided by change in time, or dI = dQ/dt. (Note: here is the simplifying assumption:) For very small changes in Q, we can assume that the voltage does not change, so that the current is constant in that interval. Therefore dV=(1/C)dQ = idt = (V/C*R)dt; rewriting this as a differential equation gives

dV/dt = V/(C*R),

which describes the discharge of a capacitor into a resistor. This is a particularly simple example, and differential equations of more complex processes can get very messy, and many cannot be solved explicitly.

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