Building a fundamental solution system one differential equation with a parameter

Many physical processes are described by problems for evolutionary differential equations. When studying such problems, the behavior of their solutions over a long time is of great interest, since it shows what the process described by the problem evolves to. It is more preferable to study the behavior of solutions to evolutionary problems with long time using asymptotic methods than numerical ones, since usually the modulus of the difference between the true solution and the numerical solution is estimated from above by a value proportional to the length of the interval on which the numerical method is applied. It is known that the study of problems for evolutionary differential equations can be reduced to the study of problems for differential equations with a parameter, while the behavior of solutions to problems for evolutionary equations for a long time will be determined by how the solutions of problems with a parameter depend on the parameter. The paper considers one homogeneous ordinary differential equation with a variable coefficient and a complex parameter, to the study of which a large class of problems for evolutionary differential equations can be reduced. The functions forming the fundamental system of solutions of the considered equation are explicitly constructed. The obtained representations allow us to trace the dependence of the constructed functions on the parameter.