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!