A General Form for Solvable Linear Time Varying Singular Systems of Differential Equations

A canonical form is derived for all linear solvable systems $E(t)x'(t) + F(t)x(t) = f(t)$ with sufficiently smooth coefficients E, F Using this form it is shown that for all smooth enough solvable systems a class of recently defined numerical imbedding methods and an algorithm to compute the manifold of consistent initial conditions always work. In addition, necessary and sufficient conditions are given on $E(t)$, $F(t)$ to insure solvability in the case when $E(t)$, $F(t)$ are infinitely differentiable.