Differential Equations

Up until now we have always solved equations that are static. Eg 2x +3 = 8 or 4x 3 -5x 2 = 0 However, nothing in the world about us is still. There is a type of equation that can express change. These equations are called Differential Equations. At least one term in a DE is a derivative or rate of change.

A differential equation (DE) connects a rate of change to another variable Slope = dy / dx Velocity = ds / dt Acceleration = dv / dt

Other DEs might look at the rate of change that occurs when dv / dc = voltage drop across capacitors dP / dt = population change over time dl / dw = change in spring length as weight increases

Common situations involving DEs include: Rate of changeDEExample Is proportional to x dy / dx = kxAs income increases the tax you pay increases Is inversely proportional to x dy / dx = k 1 / x = k / x As speed increases the time taken to complete a journey decreases Of an amount is proportional to the amount itself dy / dx = kyAs population growth increases, the population increases

We can set up a DE from a physical change if we are told how a rate of change is linked to some other quantity. Eg 1 Water is flowing out of a lake at 400litres per second. Express this as a DE Rate of change = vol of water decreasing over time dv / dt = -400 l / s

Eg 2 A population of bacteria is increasing in such a way that the rate of increase is proportional to the number of bacteria N present at time t Rate of change in number of bacteria α number of bacteria dN / dt α N dN / dt = kN