Liouville-Morducai-Boltovsky criterion

The Liouville – Mordukhai – Boltovsky criterion is a criterion for the existence of a solution in generalized quadratures of a linear homogeneous ordinary differential equation of arbitrary order.

Content

History

A special case of the criterion (for linear homogeneous equations of the second order) was proved by the French mathematician Liouville in 1839. Developing the Liouville method, the Russian mathematician Mordukhai-Boltovskaya in 1910 proved a criterion for equations of arbitrary order ^[1] :

Wording

N-th differential equation

y^{(n)}+p_{1}y^{(n-1)}+\cdots +p_{n}y=0

{\ displaystyle y ^ {(n)} + p_ {1} y ^ {(n-1)} + \ cdots + p_ {n} y = 0}

with coefficients $p_{i}$ ${\ displaystyle p_ {i}}$ from functional differential field $K$ ${\ displaystyle K}$ , all of whose elements are representable in generalized quadratures, is solved in generalized quadratures if and only if both of the following conditions are satisfied:

Firstly, it has a solution of the form

y_{1}(x)=\exp \int _{x_{0}}^{x}f(t)\,dt,

{\ displaystyle y_ {1} (x) = \ exp \ int _ {x_ {0}} ^ {x} f (t) \, dt,}

Where $f$ ${\ displaystyle f}$ Is a function lying in some algebraic extension $K_{1}$ ${\ displaystyle K_ {1}}$ fields $K$ ${\ displaystyle K}$ ,

Secondly, the differential equation of the (n − 1) th order per function $z=y'-{\frac {y_{1}'}{y_{1}}}y$ {\ displaystyle z = y '- {\ frac {y_ {1}'} {y_ {1}}} y} with odds from the field $K_{1}$ ${\ displaystyle K_ {1}}$ obtained from the original equation by the procedure of lowering the order is solved in generalized quadratures over the field $K_{1}$ ${\ displaystyle K_ {1}}$ .

Notes

↑ A. G. Khovansky . Topological Galois theory: solvability and unsolvability of equations in finite form. - M .: Publishing House MTsNMO , 2008. (p. 54-55).

Literature

A. G. Khovansky. Topological Galois theory: solvability and unsolvability of equations in finite form. - M .: Publishing House MTsNMO , 2008. - 296 p.

[1] A. G. Khovansky . Topological Galois theory: solvability and unsolvability of equations in finite form. - M .: Publishing House MTsNMO , 2008. (p. 54-55).