exact solution

简明释义

精确解

英英释义

例句

1.The physics professor demonstrated how to derive the exact solution of the wave equation.

物理教授演示了如何推导波动方程的精确解。

2.The mathematician was able to find the exact solution for the complex equation.

这位数学家能够找到复杂方程的精确解。

3.The software provided an exact solution to the optimization problem.

该软件为优化问题提供了一个精确解。

4.In engineering, sometimes an exact solution is necessary to ensure safety.

在工程中，有时需要精确解以确保安全。

5.Finding the exact solution can be time-consuming but rewarding in research.

在研究中，找到精确解可能耗时但值得。

作文

In the field of mathematics, finding an exact solution (确切解) to a problem often represents the ultimate goal of any analysis. An exact solution is one that is precise and fully satisfies the conditions of a given equation or scenario without any approximations. This concept is crucial in various branches of mathematics, including algebra, calculus, and differential equations. The pursuit of an exact solution can be both rewarding and challenging, as many mathematical problems do not have straightforward answers. For instance, consider the quadratic equation ax² + bx + c = 0. The exact solution can be derived using the well-known quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). This formula gives us the precise values of x that satisfy the equation, demonstrating how an exact solution can provide clarity in what might otherwise be a complex situation. However, not all mathematical problems yield such neat exact solutions. In many cases, especially in real-world applications, we encounter situations where an exact solution is either too complicated to compute or simply does not exist. For example, when dealing with non-linear equations or systems of equations with multiple variables, mathematicians often resort to numerical methods or approximations to find a solution that is 'good enough' for practical purposes. These approximations can be very useful, but they lack the precision that an exact solution offers. Moreover, the quest for an exact solution extends beyond pure mathematics into fields such as physics, engineering, and economics. In physics, for instance, the equations governing motion, energy, and waves often have exact solutions that can be derived under ideal conditions. However, when real-world factors come into play, such as friction or air resistance, these exact solutions may no longer apply, necessitating the use of approximations or simulations. In engineering, the design of structures and systems often relies on mathematical models that aim to produce exact solutions for stress, strain, and other physical properties. Yet, engineers must also account for uncertainties and variations in materials and loads, which can complicate the search for an exact solution. As a result, engineers frequently work with probabilistic models and simulations to ensure safety and functionality, acknowledging that while an exact solution is ideal, it may not always be attainable. In conclusion, the notion of an exact solution (确切解) is fundamental in mathematics and its applications across various disciplines. While the pursuit of an exact solution can lead to profound insights and understanding, it is essential to recognize that not all problems yield such solutions. The balance between seeking precision and embracing practicality is a key aspect of mathematical problem-solving, reminding us that while exact solutions are valuable, flexibility and adaptability often play a crucial role in addressing real-world challenges.