1. The problem statement, all variables and given/known data 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: 2. Relevant equations 3. 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 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!