## 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
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