Initial Value Linear system of DQ's

[scorpius1782]

Homework Statement

The unique solution of the linear system of differential equations
$\frac{dv}{dt}=-34v+ -16w, v(0)=-1$
$\frac{dw}{dt}=80v+ 38w, w(0)=-3$
is: (enter the smaller of the eigenvalues first, and note that all entries here are integers)
$v(t)= C_1 e^{-2t}+C_2 e^{6t}$
$w(t)= C_3 e^{-2t}+C_4 e^{6t}$

I plugged in the exponential values since they're easy to get and not my problem.
Since I already know the answer to this practice problem:
$C_1=-11$
$C_2=10$
$C_3=22$
$C_4=-25$

Homework Equations

The Attempt at a Solution

I just can't figure out how they get the constants.

The eigenvalues are -2 and 6. And the eigenvectors are [-1,2] and [-2, 5]

I thought I was suppose to set the vectors in a matrix and set equal to the initial values but this doesn't work in anyway I've tried at all. I see that C₁+C₂=-1 and that the other two constants add up to -3 but I have no clue how they picked out those numbers. The example we did in class only had 1 constraint and was an annoyingly simple problem.

I've done everything I can think of to extract the method but am just missing the method. I'm sure it will be very simple. If anyone can please help me I'd appreciate it.

 

[scorpius1782]

Nevermind, solved it.