Consider the initial value problem dx _ = Ax, with jc(0) = xo, dt where A is an upper triangular nxn matrix with m distinct diagonal entries k \........km. See the examples in Exercises 45 and 46. a. Show that this problem has a unique solution x(t)% whose components x/ (t) are of the formXi(t) = p](t)eXl' H-------- 1- Pm(t)eKm',for some polynomials pj(t). Hint: Find first xn(t), then x-\ (/), and so on. b. Show that the zero state is a stable equilibrium solution of this system if (and only if) the real part of all the Xj is negative.
1. Plane Analytic Geometry 1.1.The Cartesian System: numbering system used to identify points on a 2 dimensional plane. If 2 points are on the same line they are considered I I II IV collinear. Theorem 1.1: The distance between two points P & P i1 equ2l to √ (x2-x1) +( y 2y 1 2 1.2.If 3 points (A, B, & P) lie on the same line and P is a point on line AB the coordinates of P will be: x= x +r(x 1x ) 2 1 & y=y +1(y +y2) 1 Where r= the proportional distance from A to B point. If P is the midpoint AB, then the coordinates