Methods for Deriving Linearly Independent Solutions of the Differential Equation with Repeated Roots

Abstract

When the eigenvalue \(\lambda\) of a higher order homogeneous linear differential equation with constant coecients is the repeated root of multiplicity \(k\), the differential equation has exactly \(k\) linearly independent solutions. Different textbooks often use different ways to deal with this part of the content. "Advanced Mathematics" by Tongji University and "Ordinary Differential Equations" by Sun Yat-sen University are two commonly used textbooks for science and engineering majors and mathematics majors of universities in China. The former directly gives the conclusion, while the reasoning skills of the latter are not easily understood and mastered by many students, which leading to the degeneration of the mastery of this part into "knowing the conclusion" and "being able to use the conclusion". This is contrary to the principle that knowledge is only a carrier and teaching must focuses on ability cultivation. We discuss two new methods based on operator decomposition and the solving method for first order linear differential equation. This method is easier to understand and grasp, and can be processed in the same way for real and complex eigenvalues.