Please help with ODE!!

Question

Use the Eigenvalue Method to solve the following system of ODEs:



  

My answer was λ³ +3λ²-4=0

Eigenvalues: 1, eigenvector: 1, -1, 1

Eigenvalues: 2,eigenvector: -1, 1, 0

Answer 1

Your equation is right, only igenvalue not 2 λ but λ = - 2 of multiplicity 2, but igenvector (- 1, 1, 0) right and for igenvalue λ = 1 igenvector is (1, - 1,1).

Now we have 1 independent solution: (1, - 1, 1)e^t and second independent solution: (-1, 1, 0)e^-2t .

To get 3rd independent solution: (A + 2E)(x, y, z) = (-1, 1, 0); 4x + 4y + 3z = -1, - 4x - 4y - 3z = 1, x + y + z = 0

x = 0, y = - 1/4, z = 1/4. We get one more igenvector: (0, -1, 1).

Now solution:x(t) = C₁(1,- 1, 1)e^t + C₂(- 1, 1, 0)e^-2t + C₃[(- 1, 1, 0)t + (0, - 1, 1)]e^-2t_.

Here rows mean columns