A={{0,1},{1,2}}

Eigenvalues[A]

l1 = 1 + Sqrt[2]

l2=1-Sqrt[2]

Eigenvectors[A]

V={-1+Sqrt[2],1}

W={-1-Sqrt[2],1}

W*l2==A.W

y = A *V*Exp[l1*t] +B*W*Exp[l2*t]

dy = D[y,t]

ddy = D[dy,t]

ddy-y-2*dy

Simplify[%]