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The "Takeuchi method" is a new approach for solving differential equations. It had been invented by Yu Takeuchi by using an algebraic method and geometric sketch of the solution, all in one direction. This article will explain this new approach and describe what makes it work better than standard methods like Gaussian elimination and the successive over-relaxation (SOR) algorithm. This process can be used with any mathematical software such as MATLAB, Maple, or Mathematica. We'll also discuss how to use this method with Wolfram Alpha to calculate approximations for numerical purposes. This article will give you a more complete understanding of the "Takeuchi Method". We first need to understand the idea of "the method of sketches" proposed by Leonardo da Vinci. Leonardo had applied this method to geometry, painting, clockmaking, and even military strategies. His sketches were so brilliant that at times people thought he used magic or superpowers, but he really just had a better way of looking at things. Here are some examples of his work:He would start by sketching the problem on paper with various ideas ("poses" or "studies") until he found the best solution. This sketch would be compared with other sketches and ideas to see which was the best. This method is based on observation and practical problem solving. Leonardo's sketches were so good that sometime they would work just as well as the final solutions without being tested or revised. In other words, all he used was his intuition to create something that was amazing. In geometry, the foundations of math, there are two main ways to find a solution: one-sided and two-sided methods. One-sided methods are fast because they do not need a lot of calculations. But they are not very precise because the calculations are done on one side of the problem. Two-sided methods can be very precise but need many calculations to find a solution. On average, one-sided methods are about 80% inaccurate while two-sided methods are only about 65% accurate. The Takeuchi method is based on that information and is closer to finding solutions than other techniques. Takeuchi used his artistic brain to create sketches that would be more precise than any other technique based on one-sided methods. A "sketch" is what Leonardo used when sketching geometry problems in his book De Geometria, or when solving these types of problems with an abacus or calculator with the slide rule. In this sense, a "sketch" is a fast way of finding a solution close to the actual result, and if it looks right on paper, it will probably work. Takeuchi's method is original and novel, but many people had thought about the same thing before him. For example, Newton and Leibniz had tried to develop math that would be both precise and fast. Today we call this "symbolic methods". Symbolic methods use letters to represent variables instead of numbers. They use algebraic expressions to represent functions instead of tables or graphs. cfa1e77820