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*eλ2*t*v1 + b*eλ2*t*v2 + c*eλ3*t*v3 ???
I don't understand why, or what this means.