A=(({{-4, 2}, {2, -1}}))

{v1[t], v2[t]} /. DSolve[
v1'[t]==-4v1[t]+2v2[t]+(t^(-1)) && v2'[t]==2v1[t]-v2[t]+2(t^(-1))+4,
{v1[t], v2[t]}, t
] // Expand // Flatten// MatrixForm

JordanDecomposition[A]

DD=(({{0, 0}, {0, -5}})); CC=(({{1, -2}, {2, 1}})); (CCinv=Inverse[CC]) // MatrixForm

CCinv.{(t^(-1)), 2(t^(-1))+4} // Simplify

CC.{Log[t]+(8/5)t+(Subscript[C, 1]), (4/25)+(Subscript[C, 2])(E^(-5t))} // Simplify // ColumnForm

Ψ=(({{1, -2(E^(-5t))}, {2, (E^(-5t))}})); Inverse[Ψ] // MatrixForm

Inverse[Ψ].{(t^(-1)), 2(t^(-1))+4} // Simplify

Integrate[Inverse[Ψ].{(t^(-1)), 2(t^(-1))+4}, t]

Ψ.Integrate[Inverse[Ψ].{(t^(-1)), 2(t^(-1))+4}, t] // Expand