G. Floquet studied linear differential equations with periodic coefficients. He introduced the concept of a fundamental system of solutions for such equations, which are periodic functions with a specific multiplier. The key idea is that the solutions can be expressed in terms of polynomials in x with coefficients that are periodic functions of the same period and multiplier. Floquet derived an equation, called the fundamental equation, whose roots determine the properties of the solutions. This equation has degree m (the order of the differential equation) and its roots are related to the periodicity and multiplier of the solutions. Floquet showed that the number of linearly independent periodic solutions of second kind is determined by the multiplicities of the roots of the fundamental equation. He also demonstrated that the fundamental system can be constructed by considering the periodicity of the solutions and their transformations under shifts of the variable. The solutions can be expressed in a specific analytical form involving polynomials in x with periodic coefficients, which is a characteristic feature of linear differential equations with periodic coefficients. The work establishes a fundamental relationship between the periodicity of solutions and the structure of the differential equation.G. Floquet studied linear differential equations with periodic coefficients. He introduced the concept of a fundamental system of solutions for such equations, which are periodic functions with a specific multiplier. The key idea is that the solutions can be expressed in terms of polynomials in x with coefficients that are periodic functions of the same period and multiplier. Floquet derived an equation, called the fundamental equation, whose roots determine the properties of the solutions. This equation has degree m (the order of the differential equation) and its roots are related to the periodicity and multiplier of the solutions. Floquet showed that the number of linearly independent periodic solutions of second kind is determined by the multiplicities of the roots of the fundamental equation. He also demonstrated that the fundamental system can be constructed by considering the periodicity of the solutions and their transformations under shifts of the variable. The solutions can be expressed in a specific analytical form involving polynomials in x with periodic coefficients, which is a characteristic feature of linear differential equations with periodic coefficients. The work establishes a fundamental relationship between the periodicity of solutions and the structure of the differential equation.