By introducing the vector function z(t) = x1(t) x2(t) x _ 1(t) x _ 2(t) , show that the second-order system d2x dt2 = Ax(t) in Example 8 can be expressed as a first-order system dz dt = Bz(t), where B = 0 0 1 0 0 0 0 1 3 2 0 0 2 3 0 0 . Find the eigenvalues and eigenvectors of B, calculate etB, and solve the original problem. (Hint: Part c of Exercise 5.1.9 gives a slick way to calculate the characteristic polynomial of B, but its not too hard to do so directly.)

Review Notes for Calculus I Symmetry: A graph is symmetric with respect to the y-axis if whenever (x, y) is a point on the graph then (-x, y) is also a point on the graph. Some even functions (y=x , y=x , etc.) have symmetry with respect to the y-axis. These graphs usually are parabolas (u-shaped graphs). To figure out if a graph has y-axis symmetry, then...