Abstract: A concise description is given for the growth and development of the fundamental equations for various disciplines. It is emphasized that there are two main different stages to derive solutions of fundamental equations. The first stage is based on the analytical fundamental equations; if the exact analytical solutions cannot be derived, some approximate solutions are applied instead. However, when the computer and the computational mathematics have developed rapidly, almost all derivations are done by the numerical calculation. Even so, the exact analytical solutions still have their own important meaning. The first is the theoretical value. Moreover, the usable calculating extent of the analytical solutions is commonly larger than that calculated with the concretely numerical method. The most important point is that the analytical solutions are absolute accurate, they can be used as a benchmark solution to check and judge the accuracy of the numerical calculating results, as well as the convergence, stability etc.; and they also can be used to inspire the scientists to improve their numerical computation technology, such as the difference scheme and the network strategy. Therefore, for years there still are hundreds of papers about the analytical solutions. For this discipline, Chinese scientists are in a leading position, some of their achievements are introduced briefly. In addition, a new approach to obtaining concise exact solutions is given.

Key words: Exact analytical solution, Fundamental equation, Heat engine, Method of separating variables with addition, Compliance optimization, Feasible domain adjustment, Keywords: Robust topology optimization, Perturbation load, Structure optimization