In 19–24, convert the given second-order equation into a first-order system by setting . Then find all the critical points in the -plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12). Figure 5.12 Examples of different trajectory behaviors near critical point at origin

Solution : Step 1 of 4 : In this problem, we need to convert the given second-order equation into a first-order system by setting and find the critical points, then sketch the direction fields, and describe the stability of the critical points.Step 2 of 4 : Given system of equation is y’’ + y - = 0v = Differentiate itv’ = y’’ y’’ = -y + v’ = - y