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The secret weapon of differential equations: Do you know how the method of variables originated?
In the world of mathematics, differential equations play an extremely important role. The explanations and solutions of these equations have greatly affected the progress of many fields such as physics and engineering. The method of variables, as one of the common techniques for solving non-homogeneous linear differential equations, also has its own historical origins and evolution. How exactly did this approach originate? Let’s dig into it.
The variable method, also known as the constant change method, is a solution method proposed for non-homogeneous linear ordinary differential equations. This method transforms difficult-to-solve problems into a more tractable form.
The fundamental starting point of the variable method is to find a special solution to a non-homogeneous differential equation. In some first-order non-homogeneous linear differential equations, solutions can usually be found through other simpler methods, such as the integration factor and undetermined coefficient methods. However, these methods often rely on some conjectures and are not applicable to all non-homogeneous linear differential equations. In this context, the emergence of the variable method provides mathematicians with a more extensive problem-solving tool.
Initially, the Swiss mathematician Leonhard Euler first outlined the prototype of the method of variables in the mid-18th century, and it was further developed by the Italian-French mathematician Joseph-Louis Lagrange in subsequent literature. Complete. When Euler was studying the mutual disturbance of Jupiter and Saturn, he had already begun to explore differential equations related to planetary motion; Lagrange formally applied this method to changes in planetary motion in 1766, and then from 1808 to The general form of the method of variables was finally formed in 1810.
The excellence of the method of variables lies not in its computational power, but in its versatility and flexibility, which can be effectively applied to many linear differential equations.
The basic idea of the variable method is to express the solution of the differential equation as a combination of the solution of the homogeneous equation and a certain form of function. Specifically, for a non-homogeneous linear differential equation, we will first find the basic solution of the corresponding homogeneous equation, and then use some differentiable functions to represent the special solution. Behind this approach, we reflect the intuitive thinking formed by mathematicians in their understanding of system dynamics, and it is also an important means of mathematics in describing physical phenomena.
Through this method, mathematicians can not only solve various linear differential equations, but also extend these methods to the field of partial differential equations. In physics, linear evolution equations such as heat equations and wave equations can use this method to simplify problems and obtain effective theoretical results. These so-called "special solutions" are actually derived from the fundamental understanding of physical phenomena and are an extremely successful example of the combination of mathematics and physics.
In today's mathematics education, the variable method has become one of the basic methods for learning differential equations, and mathematicians are constantly applying it to more complex problems. With the development of mathematical theory, the application of this method has become more and more abundant, covering many aspects from simple ordinary differential equations to complex partial differential equations.
The variable method is not just a technique, it represents a mathematical thinking mode that allows us to conduct effective analysis and deduction when facing complex problems.
However, the history does not end there. The development of the method of variables reflects the inseparable connection between mathematics and science. In this process, mathematicians and scientists influenced each other and jointly promoted the evolution of theory. To what extent can the method of variables influence future scientific research and the development of mathematical methodology, and how will its importance continue?
