I solved it but I need help in two parts.For first part, How to show thev formula is a solution of ODE?Second, for the third part, how to show it is bounded because I can not integratw matrix? Consider the nonhomogeneous equation,x' = Ax + g(t),for some g(t) : R → R". Assume that A is hyperbolic, i.e it has no eigenvalues on theimaginary axis. Assume that g(t) is uniformly continuous and bounded, i.e. there exists aM > 0 such that |g(t)| < M for all t. First verify thatrtx(t) = e4t P,v +T eA(t-1) Pag(7)dr - | " eA(t-r) Pug(7)dr,is a solution for any v E R". What is the initial condition? Now prove that the solution isbounded for all t > 0.

The given non homogeneous equation is x'=Ax+gt dxdt-Ax=gt Integrating factor = e∫-Adt=e-At…

Answered: Consider the nonhomogeneous equation,…

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