Why do we often utilize simplifying assumptions when we derive differential equations?

Simplifying assumptions are statements that make a particular mathematical model or problem easier to understand and work with. These assumptions usually involve reducing the number of variables or simplifying the relationships between variables. This results in a less complicated problem that remains an accurate representation of the real-world phenomenon.

There are several reasons why simplifying assumptions are frequently made while deriving differential equations: a) Complexity reduction: Most real-world problems involve countless interacting variables and factors, making it difficult to model them realistically. Simplifying assumptions help reduce the complexity of these models, making them more manageable to work with. b) Mathematical tractability: Many real-world problems are mathematically intractable, meaning they cannot be solved using known analytical methods. Simplifying assumptions can make these problems tractable, thereby allowing us to analyze them using familiar mathematical tools and techniques. c) Reduction of noise: Real-world data often contains a considerable amount of noise and random fluctuations. Simplifying assumptions can help separate the essential underlying patterns from the noise and fluctuations, aiding in identifying important relationships and dependencies. d) Easier interpretation: Simplified models typically have fewer variables and relationships, which tends to make the interpretation of their results and implications more straightforward for researchers and practitioners.

Some examples of simplifying assumptions when deriving differential equations include: a) Population growth models: The logistic growth model assumes that the growth rate of a population is directly proportional to the current population size and the remaining capacity for growth. This assumption greatly simplifies the dynamics of population growth, allowing for easier analysis and interpretation. b) Vibrating string: When analyzing the vibration of a string, it's often assumed that the string's tension remains constant and that the motion of the string is restricted to a single plane. These simplifying assumptions make it possible to derive the wave equation governing the string's motion, which can be solved analytically. c) Newton's law of cooling: When modeling the cooling of an object, it is often assumed that the object's heat capacity remains constant, and that the object is cooling within an environment with constant temperature. These simplifying assumptions make it possible to derive the differential equation for Newton's law of cooling, which can then be solved analytically for the temperature of the object as a function of time. Overall, simplifying assumptions play a crucial role in formulating and solving differential equations. They help to reduce the complexity of the problem, while still maintaining an accurate representation of the real-world phenomenon, making it easier to analyze and understand.