Solve the initial value problem?

Question

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→∞.

Answer 1

Here we have 2 distinct roots to the characteristic equations r = -1, 2 so the general solution becomes:

y(t) = C₁e^-t + C₂e^2t

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 e^2t → ∞ as t → ∞ we need to force that C₂ = 0, doing so gets you a = -2