Could you please help with question 286 and 287? Thanks

Exercise 283 Exercise 284 Exercise 285 Exercise 286 71 5.6 Differential equations Let u(t) be a vectorvalued function of the real variable t. Our objective is to nd u(t) that satises the differential equation (:1 Eng) = Au(t) + b(t), t > 0, (5-4) where A is a constant matrix and b(t) is a known vector-valued function. First some auxiliary facts. Suppose tA has all its eigenvalues inside 9 where f is analytic. Show that (1 5mm) = f'(tA)A = Af'UM- The proof is quite easy if you use a Taylor series expansion, but not general enough. In general you have to use Cauchy's formula and the fact that since the integral is absolutely converging you can differentiate inside the integral. We rst look at the homogenous equation (1 Enth) = AuhU), t > 0. Verify that a solution is uh(t) = emuhm). With a little effort one can establish that this is the only solution. One approach is to use the Jordan decomposition to reduce the problem to a set of single variable ODEs and appeal to the scalar theory. Here we take an approach via Picard iteration that also generalizes to non-constant coe'icient ODEs. Let [0, T] be the interval over which a solution to the ODE (1 51111\") = Aw), uum) = 0, exists. If we can show that uu(t) = 0 then we would have established uniqueness. [Why?). Since the derivative of uu exists, it must be continuous. Let ||u[t)| S L 1. Conclude that u(t) = 0 for t 6 [0, T]. Now that we have uniqueness, we can look at the form of the homogenous solution and guess that a particular solution of the differential equation is t ups) = / e{t_3)Ab(s)ds, 0 assuming that up(0) = O. Show that e(t+3)A = etAe3A = e3AetA. Since the exponential is an entire function an easy proof is via a Taylor series expansion for the exponential function. Show that e" = I. Show that 8A = (8A)_1. Verify that up(t) is indeed a solution of equation 5.4. Therefore the general solution to equation 5.4 is t, u(t) = A e{t_3)Ab(s)ds + emuw)