Topics In Linear Algebra and Its Applications - STEM2

Hence two linearly independent solutions are[ ]e iω 2t 11ande −iω 2t[11].Again the solution space is spanned by[1cos(ω 2 t)1]and[1sin(ω 2 t)1].Suppose the initial conditions areu 1 (0) = Adu 1 (0)= 0dtu 2 (0) = 0du 2 (0)dtLet the solution be[ ]1u = c 11 cos(ω 1 t) +c−1 12 sin(ω 1 t)[1−1= 0][1+c 21 cos(ω 2 t)1][1+c 22 sin(ω 2 t)1],where the constants are to be determined by the initial conditions.From the first initial condition we havec 11 + c 21 = AFrom the third initial condition we have−c 11 + c 21 = 036