Correct : 0.85

Given differential equation: d²x(t)/dt² + 3dx(t)/dt + 2x(t) = 0
Initial conditions: x(0) = 20 and x(1) = 10/e
We need to find x(2)

First, I'll write the characteristic equation:
r² + 3r + 2 = 0
Factoring: (r + 1)(r + 2) = 0
So the roots are r₁ = -1 and r₂ = -2

The general solution is:
x(t) = C₁e^-t + C₂e^-2t

Now applying the first condition x(0) = 20:
C₁ + C₂ = 20 ... (1)

Applying the second condition x(1) = 10/e:
C₁e^-1 + C₂e^-2 = 10/e
This simplifies to: C₁ + C₂/e = 10 ... (2)

From equation (1): C₁ = 20 - C₂
Substituting in equation (2):
(20 - C₂) + C₂/e = 10
20 - C₂ + C₂/e = 10
C₂(1/e - 1) = -10
C₂ = 10e/(e - 1)
C₁ = (10e - 20)/(e - 1)

Finally calculating x(2):
x(2) = C₁e^-2 + C₂e^-4
x(2) = [(10e - 20)/(e - 1)] × e^-2 + [10e/(e - 1)] × e^-4
x(2) = [1/(e - 1)] × [(10e - 20)e^-2 + 10e·e^-4]

Putting e = 2.718:
e - 1 = 1.718
After calculation: x(2) ≈ 0.856

Answer: x(2) = 0.856