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2. A vector with n (real-valued) components can be represented as a column vector -0 ER". Let r(t) be a time-varying vector that satisfies the differential equation (t) = Ar(t), where AERnxn is a (square) matrix with constant entries. a. Is the above differential equation a linear differential equation? Is it homogeneous? If z(0) = , is a given initial value of the vector r(t), show that r(t) is given by x(t) = u(t).xo, where (t) is an n x n matrix that is a function of time t. Find an expression for this matrix as a series in time, and show that it satisfies the differential equation d A(t)(t). dt b. Now let X(t) satisfy the differential equation: i(t) = Ar(t) + b, where be R" is a constant vector. Is the above a homogeneous equation? Show that the solution to this differential equation is given by: r(t) = $(t)xo + Ace o(t T)bdt. c. Now let z(t) satisfy the differential equation: i(t) = = A(t).x(t) + b(t). Is this an autonomous equation? Is it a linear differential equation? Is the equation linear if A(t) is replaced by A(z) where A(2) is a (non-constant) function of x? 2. A vector with n (real-valued) components can be represented as a column vector -0 ER". Let r(t) be a time-varying vector that satisfies the differential equation (t) = Ar(t), where AERnxn is a (square) matrix with constant entries. a. Is the above differential equation a linear differential equation? Is it homogeneous? If z(0) = , is a given initial value of the vector r(t), show that r(t) is given by x(t) = u(t).xo, where (t) is an n x n matrix that is a function of time t. Find an expression for this matrix as a series in time, and show that it satisfies the differential equation d A(t)(t). dt b. Now let X(t) satisfy the differential equation: i(t) = Ar(t) + b, where be R" is a constant vector. Is the above a homogeneous equation? Show that the solution to this differential equation is given by: r(t) = $(t)xo + Ace o(t T)bdt. c. Now let z(t) satisfy the differential equation: i(t) = = A(t).x(t) + b(t). Is this an autonomous equation? Is it a linear differential equation? Is the equation linear if A(t) is replaced by A(z) where A(2) is a (non-constant) function of x