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.