Differential Equations

It is an equation whose derivative is defined by some function of it's dependent and independent variables, ie \( \dot y = f(x, y) \).

An important example of a DE is \(\dot y = ay\). Which basically says that the rate of change of y is proportional to y… Sounds familiar.

\[ \frac{dy}{dx}=ay \] \[ \int \frac{dy}{y}= \int a \, dx \] \[ \ln(y) + c_1 =ax + c_2\] \[ y = e^{ax + (c_2 - c_1)} \] \[ y = Ce^{ax} \quad \text{where } C = e^{c_2 - c_1} \]

We have rediscovered the canonical exponential where growth is a > 1 and decay is a < 1.