T 8. Suppose A(t) is an n x n matrix, and consider the first-order linear vector differential equation on (to, t1] dx = A(t)x + f(t) dt where x(t), f(t) ER". Show that any scalar nth order linear differential equation dnu dn-lu du Lu(t) +an-1t) +...+ai(t) + ao(t)u(t) = f(t) dtn dtn-1 dt on [to, ti] can be converted to a first-order vector equation of the form above by setting x(t) = (u(t), u'(t),..., eln1}(t))". The fundamental solution to the homogeneous vector ODE dx = A(t)x dt is an n x n matrix valued function W(t) such that W'(t) = A(t)W(t), W(to) = 1. Taking for granted that this fundamental solution exists, show that the solution to the inhomogeneous initial value problem above, with initial condition x(to) = Xo, is given by x(t) = W(t)xo + 1 (, 5) f(s) ds with Green's function G(t, s) = W(t)W(s)-1. Remark: This vector version of the Green's function can be converted to the scalar version, thereby proving the existence of Green's functions for nth order scalar ODEs. T 8. Suppose A(t) is an n x n matrix, and consider the first-order linear vector differential equation on (to, t1] dx = A(t)x + f(t) dt where x(t), f(t) ER". Show that any scalar nth order linear differential equation dnu dn-lu du Lu(t) +an-1t) +...+ai(t) + ao(t)u(t) = f(t) dtn dtn-1 dt on [to, ti] can be converted to a first-order vector equation of the form above by setting x(t) = (u(t), u'(t),..., eln1}(t))". The fundamental solution to the homogeneous vector ODE dx = A(t)x dt is an n x n matrix valued function W(t) such that W'(t) = A(t)W(t), W(to) = 1. Taking for granted that this fundamental solution exists, show that the solution to the inhomogeneous initial value problem above, with initial condition x(to) = Xo, is given by x(t) = W(t)xo + 1 (, 5) f(s) ds with Green's function G(t, s) = W(t)W(s)-1. Remark: This vector version of the Green's function can be converted to the scalar version, thereby proving the existence of Green's functions for nth order scalar ODEs