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3. 51) n2 where A ∈ C(I, R ). 3) applies to this system. 12. Now observe that linear combinations of solutions are again solutions. Hence the set of all solutions forms a vector space. This is often referred to as superposition principle. , x0 = nj=1 δj x0,j ). Using the solutions φ(t, t0 , δj ) as columns of a matrix Π(t, t0 ) = (φ(t, t0 , δ1 ), . . , φ(t, t0 , δn )). 53) we see that there is a linear mapping x0 → φ(t, t0 , x0 ) given by φ(t, t0 , x0 ) = Π(t, t0 )x0 . 54) 56 3. Linear equations The matrix Π(t, t0 ) is called principal matrix solution and it solves the matrix valued initial value problem ˙ t0 ) = A(t)Π(t, t0 ), Π(t, Π(t0 , t0 ) = I.
N we obtain a matrix solution U (t) = (φ1 (t), . . , φn (t)). The determinant of U (t) is called Wronski determinant W (t) = det(φ1 (t), . . , φn (t)). 57) If det U (t) = 0, the matrix solution U (t) is called a fundamental matrix solution. Moreover, if U (t) is a matrix solution, so is U (t)C, where C is a constant matrix. Hence, given two fundamental matrix solutions U (t) and V (t) we always have V (t) = U (t)U (t0 )−1 V (t0 ) since a matrix solution is uniquely determined by an initial condition.
In this case the Picard iteration can be computed explicitly, producing n xn (t) = j=0 tj j A x0 . j! 36) The limit as n → ∞ is given by x(t) = lim xn (t) = exp(tA)x0 . 35), we need to understand the properties of the function exp(tA). 38) which transforms A into a simpler form U AU −1 . 1. In fact, if A is in Jordan canonical form, it is not hard to compute exp(tA). 16)), where it is not hard to see that 2 tn−1 1 t t2! . (n−1)! .. . 1 t . 2 .. 39) exp(tJ) = eαt t . 1 2!