ACTA issues

## When do $n$ functions satisfy a $k$th-order homogeneous linear differential equation?

J. M. Szucs

Acta Sci. Math. (Szeged) 70:1-2(2004), 57-63
5798/2009

 Abstract. The functions $f_{1},\ldots,f_{n}$ satisfy a $k$th-order explicit homogeneous linear differential equation with continuous coefficients on a nondegenerate interval $I$ if and only if they are $k$ times continuously differentiable and $$\mathop{\rm rank}{\bf M}_{k-1}( f_{1},\ldots,f_{n};x) =\mathop{\rm rank}{\bf M}_{k}( f_{1},\ldots,f_{n};x) =\mathop{\rm const}, x\in I,$$ where ${\bf M}_{m}( f_{1},\ldots,f_{n};x)$ is the $m$th-order Wronskian matrix of $f_{1},\ldots,f_{n}$, whose rows are the successive derivatives of $f_{1},\ldots,f_{n}$. The equation is unique if its vector of coefficients is a linear combination of the $(k-1)$-jets $\langle f_{i}(x),\ldots,f_{i}^{(k-1)}(x)\rangle$ of the $f_{i}$ for every $x\in I$. The coefficients of this unique equation are infinitely differentiable or analytic if $f_{1},\ldots,f_{n}$ are such. If $f_{1},\ldots,f_{n}$ are linearly independent, then the equation is written explicitly. The results are extended to first-order systems. AMS Subject Classification (1991): 34A30, 15A15 Keyword(s): explicit homogeneous linear differential equations with continuous coefficients, Wronskian matrices Received December 30, 2002, and in revised form June 25, 2003. (Registered under 5798/2009.)