10 Ordinary Differential Equations Theory Floquet Theory Ple

10. Ordinary Differential Equations Theory (Floquet Theory). Please provide complete and correct solution done on computer or by hand with mathematical proof/explanation to all questions. Please, emphasis on complete and correct solution. The answers will be verified. This is all the information provided. Thank you very much.

Exercise 4.2. Let A(t) be an n x n matrix over R and let F(t) E R\", for every t e R. Suppose that A and F are continuous T-periodic functions of t e R. Prove that the solution, x(t), of the initial value problem x\'(t)-A(t)x(t) + F(t), x(0) x0, is T-periodic if and only if x (G

Solution

Let, x(t) be T periodicThen, x(0)=x(T)

Let, x(0)=x(T)=x0

x\'(t+T)=A(t+T)x(t+T)+F(t+T)=A(t)x(t+T)+F(t)

x\'(t)=A(t)x(t)+F(t)

Hence,

x\'(t+T)-x\'(t)=A(t)(x(T+t)-x(t))

y\'(t)=A(t)y(t),y(0)=x(T)-x(0)=0

y(t)=x(t+T)-x(t) ,

y\'(t)=A(t)y(t),y(0)=x(T)-x(0)=0

This is a IVP so it has a unique solution with the given initial condition: y(0)=0

y(t)=0 satisfies the IVP and the initial condition

HEnce, y(t)=0 is the solution.

Hence, y(t)=0

ie x(t+T)=x(t)

ie x(t) is T-periodic