modelling-using-first-order-differential-equations

Modelling using first order differential equations

Why are differential equations used to model real-world situations?

• A differential equation is an equation that contains one or more derivatives

• Derivatives deal with rates of change, and with the way that variables change with respect to one another

• Therefore differential equations are a natural way to model real-world situations involving change

  • Most frequently in real-world situations we are interested in how things change over time, so the derivatives used will usually be with respect to time t

How do I set up a differential equation to model a situation?

• An exam question may require you to create a differential equation from information provided

• The question will provide a context from which the differential equation is to be created

• Most often this will involve the rate of change of a variable being proportional to some function of the variable

  • For example, the rate of change of a population of bacteria, P, at a particular time may be proportional to the size of the population at that time

• The expression ‘rate of’ (‘rate of change of…’, ‘rate of growth of…’, etc.) in a modelling question is a strong hint that a differential equation is needed, involving derivatives with respect to time t

  • So with the bacteria example above, the equation will involve the derivative

• Recall the basic equation of proportionality

  • If y is proportional to x, then y = kx for some constant of proportionality k

    • So for the bacteria example above the differential equation needed would be 

  • The precise value of k will generally not be known at the start, but will need to be found as part of the process of solving the differential equation

  • It can often be useful to assume that k > 0 when setting up your equation

    • In this case, –k will be used in the differential equation in situations where the rate of change is expected to be negative

    • So in the bacteria example, if it were known that the population of bacteria was decreasing, then the equation could instead be written 

• Often scenarios will involves multiple things than affect the rate

  • If something causes the variable to increase then that term will be added to the rate of change

    • Such as water flowing into a space

  • If something causes the variable to decrease then that term will be subtracted from the rate of change

    • Such as water flowing out of space