X(o) = -7 0 0 0 1-3 40 4 0 1 0 -5 0 xo) 4 l 2 1 4 -1 x = points) use theorem i on page 350 to solve the above system of differential equations (see section 5.6 video). 2. (33points) use your solution to show that your solution solves the original system of differential equations.
350 chapter 5 linear systems of differential equations fundamental matrix solutions because the column vector (1) of the fundamental matrix in (2) satisfies the differential equation x = axit follows (from the definition of matrix multi- plication) that the matrix x= (/) itself satisfies the matrix differential equation x = ax. because its column vectors are linearly independent, it also follows that the fundamental matrix () is nonsingular, and therefore has an inverse matrix (0) conversely, any nonsingular matrix solution ) of eq. (1) has linearly independent column vectors that satisfy eq. (1), so () is a fundamental matrix for the system in (1). in terms of the fundamental matrix () in (2), the general solution (3) x(1) = 2x: (0) + 0x2(0)++. (1) of the system x = ax can be written in the form x(i) = (i) (4) sa where cele is an arbitrary constant vector. i is any other fundamental matrix for (1), then each column vector of ) is a linear combination of the column vectors of ), so it follows from eq. (4) that o = c (4) for some x xn matrix c of constants. in order that the solution x(i) in (3) satisfy a given initial condition . k(0) = a, it suffices that the coefficient vector e in (4) be such that (o) = xo: that is, that c= (0)'x when we substitute (6) in eq. (4). we get the conclusion of the following theorem. theorem i fundamental matrix solutions letu) be a fundamental matrix for the homogeneous linear system x = then the (unique solution of the initial value problem x x = axx(o) = x is given by x) = (0) 'xo. section 5.2 tells us how to find a fundamental matrix for the system with constant coefficient matrix a, at least in the case where a has a com- plete set of linearly independent eigenvectors vi. y. associated with the