What's the difference between differential equations and traditional mathematics?

Differential equations are significantly different from traditional mathematics in many aspects.

First of all, differential equation is a mathematical tool to describe the process of change, while traditional mathematics pays more attention to static and deterministic problems. Differential equations usually involve unknown functions and their derivatives, describing the changes of variables with time or other factors. In contrast, the equations in traditional mathematics are usually about the relationship between known quantities, and do not involve time or other dynamic factors.

Secondly, differential equations usually need to solve analytic or numerical solutions of unknown functions, while equations in traditional mathematics can be solved by algebraic operation or geometric methods. The solution of differential equations is often complicated, and it is necessary to consider the initial conditions, boundary conditions and the existence and uniqueness of possible solutions. The solution of equations in traditional mathematics is relatively simple, and usually only basic algebraic or geometric principles are needed to get the answer.

In addition, differential equations involve many special concepts and methods, such as ordinary differential equations, partial differential equations and linear differential equations. These concepts and methods are not common in traditional mathematics, and they provide powerful tools and frameworks for solving practical problems. For example, ordinary differential equations can be used to describe the trajectory of objects, current and voltage in circuits, etc. Partial differential equations can be used to describe problems in fluid mechanics, electromagnetism and other fields.

Finally, differential equations are widely used, covering natural science, engineering technology, social science and other fields. It can not only be used to solve specific mathematical problems, but also help us understand the laws and mechanisms in nature and social phenomena. In contrast, the application scope of traditional mathematics is relatively narrow, mainly limited to abstract mathematical theories and proofs.

To sum up, there are significant differences between differential equations and traditional mathematics in research object, solution method, conceptual framework and application scope. As a powerful mathematical tool, differential equations provide important support for us to understand and solve problems in the real world.