Alternative fundamental systems of solutions of the differential equation D n + a n − 1 D n − 1 + ⋯ + a 1 D + a 0 m y = 0 ${\left({D}^{n}+{a}_{n-1}{D}^{n-1}+\cdots +{a}_{1}D+{a}_{0}\right)}^{m}y=0$ , a 0 ≠ 0

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 ${\left({D}^{n}+{a}_{n-1}{D}^{n-1}+\cdots +{a}_{1}D+{a}_{0}\right)}^{m}y=0$ , a 0 ≠ 0, which are created using higher derivatives of the so-called fundamental function f ̃ $\tilde {f}$ . 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 ${\left\{{D}^{\nu }\tilde {f}\right\}}_{\nu =k}^{mn+k-1}$ , 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.