Find a fundamental matrix for x'=(3, 2, -2, -2)x. (2x2 matrix, 3 and 2 on the left, -2 and -2 on the right.)

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# 1 Answer

I answered this question yesterday, but it is no longer there. Here it is again:

The fundamental matrix of a system of homogeneous differential equations is the matrix whose columns are independent fundamental solutions of the system. Recall that each fundamental solution is of the form

**x**=

**s**e

^{rt},

where r is an eigenvalue of the original matrix and

**s**the corresponding eigenvector.For your matrix, the eigenvalues are -1 and 2, with corresponding eigenvectors (1,2) and (2,1). Therefore, a fundamental solution set is

{ (1,2) e

^{-t}, (2,1) e^{2t}}Therefore, the fundamental matrix, using your notation, is

( e

^{-t}, 2e^{-t}, 2e^{2t}, e^{2t}).
## Comments

^{-t}, 2e^{-t}, 2e^{2t}, e^{2t}). This is a 2x2 matrix, e^{-t}and 2e^{-t}is on the left, 2e^{2t}and e^{2t}on the right.Comment