Abstract: Taking the solution of the eigenvalue problem of the Legendre equation as an example, this paper shows the simple scientific idea that "the solution to a simple problem is often the basis for solving a complex problem, and complex problems can often be transformed into simple problems to solve". On the basis of the relatively simple eigenvalue problem of the Legendre equation, the solution of the associated Legendre equation eigenvalue problem, the associated Legendre polynomial, is constructed, avoiding knowledge points that may exceed the class. Call for more demonstration of the critical thinking process when attempting to solve problems in future textbook development, which should be beneficial for cultivating students ability to solve problems from the perspective of basic methodology.

Key words: eigenvalue problem, methodology, Legendre equation, Associated Legendre equation