Abstract The article deals with the issue of the existence of fundamental systems of solutions to the differential equations: D n + a n − 1 D n − 1 + ⋯ + a 1 D + a 0 m y = 0 ({D^n+a₍-₁D^n-1+ +a₁D+a₀) }^my=0, a 0 ≠ 0, which are created using higher derivatives of the so-called fundamental function f ̃ f. The problem is solved by applying a suitable transformation to the matrix appearing in the Wronskian and subsequently calculating the determinant of the modified Wronskian matrix V using Laplace’s generalized expansion. The matrix V possesses several notable properties: it is a symmetric matrix, block symmetric, and each block is itself a symmetric matrix. When applied to the differential equation, zero blocks appear below the block anti-diagonal. It is proven that the sequence of derivatives D ν f ̃ ν = k m n + k − 1 \{{D^ f\}} =₊^mn+k-1, k = 0, 1, 2, …, forms a fundamental systems of solutions to the aforementioned differential equation. Additionally, the article generalizes certain results from the author’s previous works in Mathematica Slovaca.
Jozef Fecenko (Wed,) studied this question.