Solve Diff. Equation System: Find Mistake

[prehisto]

Homework Statement

System1:
dx/dt=x+y
dy/dt=8x-y

Homework Equations

The Attempt at a Solution

detreminant=(1-λ)(-1-λ)=(λ-3)(λ+3);λ$_{1}$=-3 and λ$_{2}$=3

So system 2:
(1-λ)$\alpha$+$\beta$=0
8$\alpha$+(-1-λ)$\beta$=0

When i put λ$_{1}$=-3 in system 2 -> $\alpha$ and $\beta$=0.
the same goes for λ$_{2}$

That menas that solution in form of y=C_1*$\beta$_1*exp(λ$_{1}$*t)+C_2*$\beta$_2*exp(λ$_{2}$*t) is equal to 0. Thats wrong.

Where is my mistake?

 

[tiny-tim]

hi prehisto!

so far so good! …

(1-λ)$\alpha$+$\beta$=0
8$\alpha$+(-1-λ)$\beta$=0

now solve either line to get β = 2α, so your eigenvector is any multiple of x + 2y

 

[prehisto]

ok,that means that i can chose α₁=1 β₁=2 and
α₂=1 β₂=-4

y=C₁*β₁*exp(λ1*t)+C₂*β₂*exp(λ2*t)
x=C₁*α₁*exp(λ1*t)+C₂*α₂*exp(λ2*t)
Is this form of solution correct or I have to use something else?

 

[tiny-tim]

i think it would be better if you checked by starting with the eigenvector equations

x + 2y = Ae^3t
x - 4y = Ae^-3t