Step 1

The characteristic polynomial z² - z - 2 has two real roots, z = -1 and z = 2.

Step 2

The general solution is y = Ae^-t + Be^2t.

Step 3

The initial value y(0) = 5 gives the equation 5 = A + B.

To get a second equation, we differentiate the general solution and substitute the initial value for y'(0).

y' = -Ae^-t + 2Be^2t, 0 = -A + 2B.

The solution of the two equations for A and B is

The particular solution of the initial value problem, shown in Figure 14.6.1, is