Constructions of the general solution for a system of partial differential equations with variable coefficients (Q1192849)

The authors prove: Let \(M\) be a linear space. Let \(F_ 0(M)\) be operators \(A\) from \(M\) into \(M\) with \(A(0)=O\) and \(L(M)\) be linear operators in \(F_ 0(M)\). Let \(B\in L(M)\) and \(A\in F_ 0(M)\), and let there exist \(C\), \(D\in F_ 0(M)\) such that \(AC=BD\), then the general solution of \(Au=Bv\) satisfying the condition \(u\in C(M)\) is \(u=C(\varphi)\), \(v=D(\varphi)+\psi\) where \(\varphi\in M\), \(\psi\in\{\psi\in M; B\psi=0\}\equiv\text{Ker} B\). If \(A,C,D\in L(M)\) and \(D(\text{Ker} C)=\text{Ker} B\), then the general solution of \(Au=Bv\) satisfying the condition \(u\in C(M)\) is \(u=C(\varphi)\), \(v=D(\varphi)\), \(\varphi\in M\). Applying this assertion, the authors prove: Letting \(A\), \(B\) be linear partial differential operators with sufficiently smooth coefficients, there exist two linear partial differential operators \(C\) and \(D\) with sufficiently smooth coefficients satisfying \(AC=BD\).