IB DP Maths: AI HL复习笔记5.6.1 Modelling with Differential Equations
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
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- 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
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Worked Example
a)In a particular pond, the rate of change of the area covered by algae, A, at any time t is directly proportional to the square root of the area covered by algae at that time. Write down a differential equation to model this situation.
b)Newton’s Law of Cooling states that the rate of change of the temperature of an object, T, at any time t is proportional to the difference between the temperature of the object and the ambient temperature of its surroundings, Ta , at that time. Assuming that the object starts off warmer than its surroundings, write down the differential equation implied by Newton’s Law of Cooling.
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