Consider the initial value problem for the system of first-order differential equations
y_1' = -2y_2+1, y_1(0)=2
y_2' = -8y_1+2, y_2(0)=-1
If the matrix
[ 0 -2
-8 0 ]
has eigenvalues and eigenvectors L_1= -4 V_1= [ 1
L_2=4 V_2= [ 1
then its solution will be:
The Attempt at a Solution
e^(-4t) +e^ (4t) from eigenvalues
multiply by respective eigenvectors and set to initial conditions gives 2 sets of equations and two unknown coefficients
I am very confident with these values being right for the coefficients, I know need to know how to use these to form a general solution.
I plug these values back in to get
Then solution given is
any help is appreciated! thanks!