## Substitution method for generalized linear differential equations.(English)Zbl 0743.34023

The paper deals with the problem: how to find to a generalized linear differential equation (1) $$dx=d[A(t)]x+df$$, where $$A,f\in BV_ n^{loc}(J)$$, such an ordinary differential equation that both equations have the same solutions. This can be done by a substitution method described in the paper which is based on the notion of a logarithmic prolongation of $$A(t)$$ along an increasing function $$v(t)$$. This enables to obtain the properties of the solutions of (1) similar to those of the linear ordinary differential equations. As an example, there are found sufficient conditions for variational stability of the zero solution of the generalized linear differential equations.

### MSC:

 34A30 Linear ordinary differential equations and systems 34D05 Asymptotic properties of solutions to ordinary differential equations 34A99 General theory for ordinary differential equations 34D99 Stability theory for ordinary differential equations
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