Lecture Slides

Consider the initial value problem u = f (u, t), u(t 0) = u 0 An mth order system of s equations can be reduced to a system of ms first order equations, by introducing new variables v 1 = u , v 2 = u ,. .. , v m−1 = u (m−1). The explicit time-dependency can be removed by introducing a new variable w = t with the trivial equation w = 1, to obtain a new autonomous system of equations A linear system of ODEs has the form u = A(t)u + g(t), A(t) ∈ R s×s , g(t) ∈ R s If A does not depend on t, the system is a constant coefficient linear system u = Au + g(t) for a constant matrix A ∈ R s×s. For g(t) = 0 the equation is homogeneous with solution u(t) = e A(t−t0) u 0 where the matrix exponential for diagonalizable A = V ΛV −1 is