I got you bro :)
We are given the system of differential equations:
dx/dt = 2x + 8y
dy/dt = -x + y
We can solve this system using matrix methods. First, we can rewrite the system in matrix form:
d/dt [x; y] = [2 8; -1 1] [x; y]
The matrix [2 8; -1 1] is called the coefficient matrix of the system.
Next, we can find the eigenvalues and eigenvectors of the coefficient matrix:
|2-λ 8| |-(λ-1) 8|
|-1 1-λ| = |-1 -(λ-1)|
Using the formula for the characteristic equation, we get:
(2-λ)(1-λ) + 8 = 0
Simplifying, we get:
λ^2 - 3λ + 10 = 0
Solving for λ using the quadratic formula, we get:
λ = (3 ± sqrt(29)i)/2
Since the eigenvalues have a non-zero imaginary part, the system has complex eigenvectors. The eigenvectors can be found using the formula:
(A - λI)v = 0
where A is the coefficient matrix, λ is an eigenvalue, and v is the eigenvector corresponding to λ. Solving this equation, we get:
v1 = (1 + sqrt(29)i)/4
v2 = (1 - sqrt(29)i)/4
Using the eigenvectors, we can diagonalize the coefficient matrix:
[2 8; -1 1] = [v1 v2] [λ1 0; 0 λ2] [v1 v2]^-1
where λ1 and λ2 are the eigenvalues and [v1 v2]^-1 is the inverse of the matrix [v1 v2].
Plugging in the values, we get:
[2 8; -1 1] = [(-sqrt(29)i/4) (sqrt(29)i/4)] [(3+isqrt(29))/4 0; 0 (3-isqrt(29))/4] [(-sqrt(29)i/4) (sqrt(29)i/4)]^-1
Simplifying, we get:
[2 8; -1 1] = [(sqrt(29)i/2) (sqrt(29)i/2)] [(1+isqrt(29))/2 0; 0 (1-isqrt(29))/2] [(sqrt(29)i/2) (sqrt(29)i/2)]^-1
Now we can solve the system of differential equations using the diagonalized matrix. We let:
[x; y] = [c1 v1; c2 v2]
where c1 and c2 are constants to be determined. Substituting into the matrix equation, we get:
d/dt [c1 v1; c2 v2] = [(1+isqrt(29))/2 0; 0 (1-isqrt(29))/2] [c1 v1; c2 v2]
Expanding, we get:
c1' v1 + c2' v2 = (1+isqrt(29))/2 c1 v1 + (1-isqrt(29))/2 c2 v2
c1' v1 + c2' v2 = (1+isqrt(29))/2 c1 v1 + (1-isqrt(29))/2 c2 v2
Solving for c1' and c2', we get:
c1' = (