Figure 1: Solution curves to y 00 − 2y 0 − 3y = 0. Red solves y(0) = 1, y 0 (0) =
−1, Blue solves y(0) = 2, y 0 (0) = 3, Yellow solves y(0) = 0, y 0 (0) = 0, and
Green solves y(0) = 0, y 0 (0) = −3. With second order equations, solution
curves may indeed cross (multiple times) in the direction field.
1