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.

$$\frac{dy}{dx}=ay$$ $$\int \frac{dy}{y}=\int adx$$ $$\ln (y)+c_1=ax+c_2$$ $$y=e^{ax+(c_2-c_1)}$$ $$y=C{e}^{ax}\text{where }C=e^{c_2-c_1}$$

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