Modelling with Differential Equations

Modelling Real-world Problems

• Differential equations can be used to model a wide variety of situations in physics, engineering, biology, economics and many other fields.
• The process usually involves translating a physical situation and its condition into a mathematical language. The aspects of the situation to be considered are the dependent and independent variables, along with any parameters.
• The construction of a mathematical model often necessitates assumptions and simplifications that allow complex, real-world problems to be tackled.
• While formulating a model, variables and parameters must be carefully defined and their units of measurement noted.

Solving a Mathematical Model

• Once constructed, a mathematical model can be solved using techniques from calculus, especially differentiation and integration.
• The solution to the model provides a quantitative prediction about the system being modelled.
• When dealing with first-order linear differential equations, the integrating factor method is particularly useful for finding solutions.

Interpreting the Solutions

• Understanding the interpretation of solutions is key. The solutions could represent anything: population levels in an ecology model, the concentration of a drug in the bloodstream in a pharmacokinetic model, etc.
• Sketching the solutions in a phase plane or a direction field can provide a broad understanding of the system’s behaviour.
• It’s essential to note that the model is only a simplified representation of the real-world problem. Therefore, its predictions should be analyzed and compared with observed data to check the model’s validity and accuracy.

Modifying the Model

• Where discrepancies occur between a model’s prediction and the real-world data, the model may need to be adjusted or refined.
• This may involve introducing new variables and parameters, changing the form of the differential equation, or making more complex assumptions.
• Always keep in mind, mathematical modelling is an iterative process. The aim is to continuously improve the model to get a more accurate description of the real-world problem.

Remember, modelling with differential equations not only develops mathematical skills but also promotes critical thinking, problem-solving ability, and broadens your understanding of how mathematics can be applied to real-world situations!