Denote h1=[2 2]T the eigenvector for λ=1, then f1(t)=h1et is a particular solution. Another(linearly independent) solution can be found as the following: find vector h2 that is a solution of linear algebraic system of equations (A-1*Id)h2=h1. Here A is the matrix of your system, Id is the identity matrix. h2=[0 1]T is a such vector. Then f2(t)=(h2+th1)et is the second linearly independent solution, and the general solution is C1f1(t) + C2 f2(t), i. e.