×

Covariant constructions in the theory of linear differential equations. (English) Zbl 0597.34005

For \(n\geq 2\), let \(AD_ n\) denote the set of all nth order linear differential equations of the form \(P_ n:y^{(n)}+p_{n-2}(x)y^{(n- 2)}+...+p_ 0(x)y=0\) whose real coefficients are expressible by power series convergent on the whole real line. Consider a transformation h, \(h\in C^{n+1}(J)\), dh(t)/dt\(\neq 0\) on J, of the equation \(P_ n\in AD_ n\) into an equation \(Q_ n\in AD_ n\) given by the formula \(z_ i(t)=| dh(t)/dt|^{(1-n)/2}y_ i(h(t)),\) \(i=1,...,n\) where \(y_ i\) and \(z_ i\) are n-tuples of linearly independent solutions of \(P_ n\) and \(Q_ n\), respectively; write \(Q_ n=h(P_ n)\). Let \(u_ 1\) and \(u_ 2\) be two linearly independent solutions of an equation from \(AD_ 2\) \(p:u''+p(x)u=0.\) By definition the iterative nth order equation \(F_ n(p)\) to the equation p has the following n-tuple of solutions \(u_ 1^{n-i}(x)u_ 2^{i-1}(x),\) \(i=1,...,n\). It is proved that this construction \(F_ n\) of iterative equations is the only construction that maps \(AD_ 2\) into \(AD_ n\) and satisfies the commutative condition \(F_ n(h(p))=h(F_ n(p)).\) From the categorial point of view, \(F_ n\) is the only covariant functor of \(AD_ 2\) into \(AD_ n\) where morphisms are transformations.

MSC:

34A30 Linear ordinary differential equations and systems
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
18B99 Special categories
PDF BibTeX XML Cite
Full Text: EuDML