# Initial Value Linear system of DQ's

1. Dec 6, 2013

### scorpius1782

1. The problem statement, all variables and given/known data
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:
$C_1=-11$
$C_2=10$
$C_3=22$
$C_4=-25$

2. Relevant equations

3. 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.

2. Dec 6, 2013

### scorpius1782

Nevermind, solved it.

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