IB DP Maths: AA HL复习笔记5.10.3 Modelling with 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

What is the logistic equation?

How do I solve problems that involve a logistic equation model?

• Solving the differential equation will generally involve the technique of separation of variables
  • Usually this will also involve rearranging one of the integrals using partial fractions (see the worked example below for an example)
• You will usually be given ‘boundary conditions’ specific to the context of the problem
  • For example, you may be told the initial population at time t = 0
  • These conditions will allow you to work out the exact value of any integrating constants that occur while solving the differential equation
• You will need to take account of the context of the question in answering the question or in commenting on the model used