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**Solving homogeneous linear differential equations in terms of second order linear differential equations.**
*(English)*
Zbl 0564.12022

Let \(F\) be a differential field of characteristic zero and (1) \(L(y)=0\) be an nth order homogeneous linear differential equation with coefficients in \(F\). In what case may solutions of (1) be expressed by means of the solutions of linear differential equations with lower order? In what case is an operator \(L(y)\) reducible, e.g. \(L(y)=L_1(L_2(y))\) where \(L_1,L_2\) are linear differential operators with coefficients in \(F\)? In his Galois theory of differential fields E. R. Kolchin adduces solution of such problems (see [Differential algebra and algebraic groups. New York etc.: Academic Press (1973; Zbl 0264.12102)]). In the first case the Galois group of (1) should have a normal tower of algebraic subgroups such that any quotient is either a finite group or an algebraic group with a faithful representation of degree lower than \(n\). In the second case the Galois group of (1) must be reducible as a matrix group. However, this Galois group is unknown in all interesting cases (for example, Mathieu’s equation \(y''+(a+b \cos 2z)y=0\) over the field of \(2\pi\)-periodic functions which are meromorphic in \(\mathbb{C})\).

In his previous papers the author (see, for example [Am. J. Math. 103, 661–682 (1981; Zbl 0477.12026)]) contributed to the solution of the above problems. In the present paper he obtains some similar results about an eulerian solution of (1). This results will be useful for discussion of the structure of solutions of third order homogeneous linear differential equations with coefficients in \(\mathbb{Q}(x)\), by means of computer systems like MACSYMA and REDUCE-2.

We note that the Galois group of the equation \(y'''-xy=0\) over \(\mathbb{C}(x)\) is connected (see, the reviewer, Theoretical and applied problems for differential equations and algebra, Collect. sci. Works, Kiev 1978, 71–75 (1978; Zbl 0435.12012)]) and this simplifies slightly the consideration of this example.

In his previous papers the author (see, for example [Am. J. Math. 103, 661–682 (1981; Zbl 0477.12026)]) contributed to the solution of the above problems. In the present paper he obtains some similar results about an eulerian solution of (1). This results will be useful for discussion of the structure of solutions of third order homogeneous linear differential equations with coefficients in \(\mathbb{Q}(x)\), by means of computer systems like MACSYMA and REDUCE-2.

We note that the Galois group of the equation \(y'''-xy=0\) over \(\mathbb{C}(x)\) is connected (see, the reviewer, Theoretical and applied problems for differential equations and algebra, Collect. sci. Works, Kiev 1978, 71–75 (1978; Zbl 0435.12012)]) and this simplifies slightly the consideration of this example.

Reviewer: N. V. Grigorenko

### MSC:

12H05 | Differential algebra |

12H20 | Abstract differential equations |

34G10 | Linear differential equations in abstract spaces |

68W30 | Symbolic computation and algebraic computation |