Reminder

General Solution: non-homogeneous Equation

If {X1, X2, X3,…,Xn} is a FS for
	({{X'[t], =, A[t].X[t]}})
and XP[t] solves
	({{X'[t], =, A[t].X[t], +, F[t]}})
then
	({{X[t], =XP[t]+C1 X1[t]}})+…+CN Xn[t] 
gives all solns of ({{X'[t], =, A[t].X[t], +, F[t]}}).

We know how to find the Complementary Solution.  Remember, the solution to the homogeneous problem.

The FS solution is the eigenvalue/eigenvector stuff.  Only issues are missing eigenvectors and complex eigenvalue pairs.

Undetermined Coefficients:

All we need to know how to do is compute a solution of the non-homogenous problem

There is a solution of X'=A.X+ (Subscript[F, 0]) which is a constant vector (Subscript[B, 0]).

There is a solution of X'=A.X+ t (Subscript[F, 1])+(Subscript[F, 0]) which is (Subscript[B, 0])+t (Subscript[B, 1]).

There is a solution of X'=A.X+ (e^(γ t)) (Subscript[F, 1]) which is (e^(γ t)) (Subscript[B, 0]) 
as long as γ  is not an eigenvalue of A.

There is a solution of X'=A.X+ Cos(γ t) (Subscript[F, 0])+Sin(γ t) (Subscript[F, 1]) which is Cos(γ t) (Subscript[B, 0])+Sin(γ t)(Subscript[B, 1]) 
as long as γ i is not an eigenvalue of A.

Variation Of Parameters:

A Fundamental Set of solution {(Subscript[X, 1]),(Subscript[X, 2]),…,(Subscript[X, n])} solutions of X'=A.X gives the general solution
	X(t)=(Subscript[C, 1])(Subscript[X, 1])(t)+(Subscript[C, 1])(Subscript[X, 1])(t)+…+(Subscript[C, n])(Subscript[X, n])(t).
This is
	X(t)=Φ(t).C
where Φ=(({{(Subscript[X, 1]), (Subscript[X, 2]), …, (Subscript[X, n])}})) and C=(({{(Subscript[C, 1])}, {(Subscript[C, 2])}, {⋮}, {(Subscript[C, n])}})).  
The Fundamental Matrix Φ satisfies Φ'=A.Φ

Variation of Parameters finds a solution to X'=A.X+F of the form
	(Subscript[X, p])(t)=Φ(t).U(t).
Plug into the eqn to get
	({{Φ'.U+Φ.U', =, A.Φ.U+F}})
Remember Φ'=A.Φ and reduce to get
	Φ.U'=F
Solve for U to get
	({{U(t), =, (∫_( ))((Φ[t])^(-1))F[t]dt}})
and plug back in to get 
	({{(Subscript[X, p]), =, Φ(t).}})U(t)
The formula is deceptively simple. Finding inverses is messy.

Inverse[Array[a,{2,2}] ]//MatrixForm

Ex 1: VP

Solve
X'[t]== (({{2, 1}, {0, 7}})).X[t]+(e^(7 t))(({{4}, {10}}))

Eigensystem[(({{2, 1}, {0, 7}}))]
FM[t_]:=(({{(E^(7t)), (E^(2t))}, {5 (E^(7t)), 0}}));
F[t_]:=(E^(7t)){4,10}
FM[t].Integrate[Inverse[FM[t]].F[t],t]

Check

Simplify[
{y1[t],y2[t]}/.DSolve[{
y1'[t]== 2y1[t]+1y2[t]+4 (E^(7t)),
y2'[t]== 0 y1[t]+7y2[t]+ 10 (E^(7t))
},
{y1[t],y2[t]},t][[1]]
]