Use differential equations in modelling

Introduction to Modelling with Differential Equations

Differential equations can encapsulate a wide variety of real-world phenomena. It’s common to use them while modelling systems in a broad range of fields, from physics and biology, to economics.
The rate of change of a certain quantity, often dictated by the laws of nature, is frequently expressed by a differential equation. For instance, they can capture how a population grows, or how a physical object moves under the force of gravity.

Forming Differential Equations

When creating a model using a differential equation, it’s important to identify the variables and their rates of change. The variables generally reflect quantities of interest in the system being modelled, like population size, concentration of a substance, or distance travelled.
Physical laws or biological theories are commonly used to establish these relationships between variables. For instance, Newton’s law of cooling establishes a proportional relationship between the rate of change of temperature and the difference between an object’s temperature and its surroundings.
The linking of variables and their relationships forms the basis of the differential equation which will be the mathematical model of the system.

Solving the Model

After forming the differential equation, the next step is to solve it to derive an equation which can predict behaviour within the system.
Different techniques, like separation of variables, integrating factors, exact methods, or homogeneous equations could be utilised to find the general solution.
If the differential equation is hard to solve or is non-linear, approximations may be necessary. Numerical methods such as Euler’s method or the Runge-Kutta methods can then be employed.

Interpreting the Solution

The general solution provides a family of potential solutions. To find a specific solution that corresponds to the real-world context you’re modelling, utilise initial or boundary conditions that represent known behaviours or values of the system.
Once the particular solution has been determined, it’s time to analyze and interpret it in the context of the problem. The solution allows the prediction of the behaviour of the system, and the evaluation of that system under various conditions.

Continuous Iterative Process

Modelling with differential equations is an iterative process. Always reassessing the model’s validity and making necessary adjustments based on comparison with empirical data is a crucial aspect of this process.
The solution should be checked against any known behaviours of the system and any additional data points.

Overall, modelling with differential equations requires not only calculus and algebraic skills, but also the ability to convert real-world problems into mathematical language and to translate mathematical results back into real-world interpretation.