Fundamental systems of solutions of some linear differential equations of higher order

Abstract The article deals with the issue of the existence of fundamental systems for solving some types of homogeneous linear differential equations with constant coefficients, which are created using higher derivatives of the so-called fundamental function. The first part of the article also partially expands on new information from the article [4]. Some interesting results are presented: Let f ( x ) = x n− 1 sin βx be the fundamental function of the solutions to the differential equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>D</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>β</m:mi> <m:mn>2</m:mn> </m:msup> <m:mrow> <m:msup> <m:mo stretchy="false">)</m:mo> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mi>y</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:math> $(D^2+\beta^2)^ny=0,$ then the sequence of derivatives <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mfenced open="{" close="}"> <m:mrow> <m:msup> <m:mi>D</m:mi> <m:mrow> <m:msub> <m:mi>k</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:msup> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfenced> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:mrow> </m:msubsup> </m:math> $\left\{D^{k_i}f(x)\right\}_{i=1}^{2n}$ forms a fundamental system of solutions to this differential equation if the number of distinct even-order derivatives of f is equal to the number of distinct odd-order derivatives. Another result is that the set of n arbitrary distinct derivatives of the function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:msup> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mtext>e</m:mtext> </m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mi>x</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>λ</m:mi> <m:mo>≠</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> </m:math> $f(x)=x^{n-1}{\text{e}}^{\lambda x} (\lambda \neq 0)$ forms the fundamental system of solutions to the differential equation ( D − λ ) n y = 0. While the article does not provide a comprehensive solution to the issue, it offers the potential for generalizing the presented results.