Solve the initial value problem y"-y'-2y=0, y(0)=α, y'(0)=2. Then find α so that the solution approaches zero as t→∞.
Here we have 2 distinct roots to the characteristic equations r = -1, 2 so the general solution becomes:
y(t) = C1e-t + C2e2t
plugging in our initial conditions we get:
y(0) = C1 + C2 = a
y'(0) = -C1 + 2C2 = 2
After solving this 2 X 2 linear system (preferably using the elimination method) we should get C1 = 2(a-1)/3 and C2 = (a+2)/3.
since e2t → ∞ as t → ∞ we need to force that C2 = 0, doing so gets you a = -2