Let x\' = Ax be given by the matrix A = [3 1 0 3] Compute the eigenvalues and ei

Question

Let x' = Ax be given by the matrix A = [3 1 0 3] Compute the eigenvalues and eigenvectors of A. Explain why you have found one linearly independent solution to the system, but still need to determine another. Verify through direct substitution that x2(t) = te3t[l 0]T + e3t [0 1]T is a solution to the given system x' = Ax. Show that the solution you found in (a) above and the solution x2(t) in (b) are linearly independent, and hence state the general solution to the system. Solve the IVP with the initial condition x(0) = |3 2]T.

Explanation / Answer

I am sure you are aware, multiple question posting is prohibited. I will answer the first part and give you hints on the second part. The (repeated) eigenvalue is r = 3. This eigenvalue has only one linearly independent eigenvector, v = (1, 0)^T. Since there are not two linearly independent eigenvectors corresponding to the one eigenvalue, you do not yet have enough information to get the general solution. to see that x2 = t e^(3t) (1, 0)^T + e^(3t) (0, 1) is a solution, just plug it in. Now you have two linearly independent solutions. The general solution is x(t) = c1 (1,0)^T e^(3t) + c2(t e^(3t) (1, 0)^T + e^(3t) (0, 1) ) Solve the IVP by plugging in t = 0 and solving the associated linear system for c1 and c2.

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