Remark 3

Given the polynomials $p_{jk}(D)$, a system (47) which admits solutions when the right-hand sides are indefinitely differentiable (or differentiable of a certain order) may not admit them when the right hand sides are arbitrary regulated functions: this is shown by the example (48), which has no solution when $a(t)$ is not a primitive. We shall not undertake here to seek for supplementary possibility conditions which enter when the right-hand sides are arbitrary regulated functions.