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## Homework Statement

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

2 ]

L_2=4 V_2= [ 1

-2]

then its solution will be:

## Homework Equations

## The Attempt at a Solution

[/B]

e^(-4t) +e^ (4t) from eigenvalues

multiply by respective eigenvectors and set to initial conditions gives 2 sets of equations and two unknown coefficients

2=c_1*e^(-4t)+c_2*e^(4t)

-1=c_1*2e^(-4t)+c_2*-2e^(4t)

c_1=3/4

c_2=5/4

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

y_1=3/4e^(-4t)+5/4e^(4t)

y_2=3/2e^(-4t)-5/2e^(4t)

Then solution given is

y_1(t)=5/4e^(4t)+1/2e^(-4t)+1/4,

y_2(t)=-5/2e^(4t)+e^(-4t)+1/2

any help is appreciated! thanks!