Answer to Question #245781 in Differential Equations for Nivetha

Question

Let A(t), t in I be a real matrix. Prove that (5.6.31) can also be written as Psi(t)= Phi(t) int t 0^t Psi^ T (s)b(s)ds. (i) t in I provided Psi ^ T* (t) * Phi * (t) = E; Psi(t)=( Psi^ r^"-1")^ Tint 10^t Psi^ T (s)b(s)ds, (ii) t in I, where is a fundamental matrixfor the adjoint system x^ prime =-A^ T (t)x.