Complex Eigenvalues Question

I am trying to solve this two by two matrix differential equation, but I keep getting the wrong answer. Can anyone help please and thank you in advance?


Solve the system
dx/dt=
|-2. 1|
|-5. 2| X

Give the solution in real form.
X1(t)=___________________
x2(t)=____________________

Eigenvalues:

det | −2−λ .. 1 | = 0
..... | −5 .. 2−λ |

(−2−λ)(2−λ) − (−5)(1) = 0
−(2+λ)(2−λ) + 5 = 0
−(4 − λ²) + 5 = 0
−4 + λ² + 5 = 0
λ² = −1
λ = i, λ = −i

Check out the following for solving differential equations with complex eigenvalues:
http://tutorial.math.lamar.edu/Classes/D…

Now we find eigenvectors (we only need to find 1 eigenvector)
λ = i

[ −2 .. 1 ] [ a+bi ] = i [ a+bi ]
[ −5 .. 2 ] [ c+di ] ..... [ c+di ]

−2a − 2bi + c + di = (−2a+c) + (−2b+d)i = −b + ai
−5a − 5bi + 2c + 2di = (−5a+2c) + (−5b+2d)i = −d + ci

Matching real and imaginary parts, we get:
−2a + c = −b
−2b + d = a
−5a + 2c = −d
−5b + 2d = c

Solving, we get:
c = 2a − b
d = a + 2b

Setting a = 1, b = 0, we get c = 2, d = 1

Eigenvector:
[ .. 1 .. ]
[ 2 + i ]

X = e^(it) [ .. 1 .. ] = (cos t + i sin t) [ .. 1 .. ] = [ cost + i sint ................................ ]
................ [ 2 + i ] ............................ [ 2 + i ] .... [ 2 cost + 2i sint + i cost + i² sint ]

X = [ cos t ............ ] + i [ sin t ............ ]
...... [ 2 cost − sint ] .... [ 2 sint + cost ]

So general solution is

x₁(t) = A cos t + B sin t
x₂(t) = A (2 cos t − sin t) + B (2 sin t + cos t) = (2A + B) cos t + (−A + 2B) sin t

—————————————————————————

Check:

x₁(t) = A cos t + B sin t
x₁'(t) = −A sin t + B cos t

x₂(t) = (2A + B) cos t + (−A + 2B) sin t
x₂'(t) = (−2A − B) sin t + (−A + 2B) cos t

From original system:

x₁' = −2 (A cos t + B sin t) + (2A + B) cos t + (−A + 2B) sin t