Study and solution of degenerate systems of ordinary differential equations by means of a change of variables
For the system \(A(t) x'(t)+ B(t) x(t)= f(t)\), \(t\in T= [\alpha, \beta]\), with \(A(t)\) an \(n\times n\)-matrix degenerate for every \(t\in T\) (\(f(t)\) and \(x(t)\) are the given and sought \(n\)-dimensional vectors and \(B(t)\) is not full rank in general) the Cauchy problem \(x(\alpha)= a\) is studied. Only such degenerate systems whose solutions are independent of arbitrary vector-functions are considered (i.e. they have Cauchy-type solutions). Necessary and sufficient conditions for existence of the right resolving operator and its decomposition into a composition of first-order linear differential operators is established as well as some equivalence between the right and the left resolving operators. An algorithm for finding a resolving change of variables is proposed.
scientific article; zbMATH DE number 917166
scientific article