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## Homework Statement

Find the general solution to the following differential equations

y'1 = -12y1 + 13y2 +10y3

y'2 = 4y1 - 3y2 - 4y3

y'3 = -21y1 +21y2 +19y3

## The Attempt at a Solution

I'm a little unsure about what to do at the end, or what form to put it in.

The eigenvalues are

λ1 = 5, λ2 = -2, λ3 = 1

and the eigenvectors are

v1 = [1, -1, 3], v2 = [1, 0, 1], v3 = [1, 1, 0] (respectively)

so once I have that, what do I do? I need to put them into an exponential form like

a*e

^{λ2*t}*v1 + b*e

^{λ2*t}*v2 + c*e

^{λ3*t}*v3 ?

I don't understand why, or what this means.