In matrix form, the ODE is given by

a. Move  to the left side and multiply both sides by the integrating factor, the matrix exponential of
 to the left side and multiply both sides by the integrating factor, the matrix exponential of  ,
,  :
:

Condense the left side as the derivative of a product:

Integrate both sides and multipy by  to solve for
 to solve for  :
:

Finding  requires that we diagonalize
 requires that we diagonalize  .
.
 has eigenvalues 4 and 9, with corresponding eigenvectors
 has eigenvalues 4 and 9, with corresponding eigenvectors  and
 and  (explanation for this in part (b)), so we have
 (explanation for this in part (b)), so we have





b. Find the eigenvalues of  :
:


Let  and
 and  be the corresponding eigenvectors.
 be the corresponding eigenvectors.
For  , we have
, we have

which means we can pick  and
 and  .
.
For  , we have
, we have

so we pick  .
.
Then the characteristic solution to the system is


c. Now we find the particular solution with undetermined coefficients.
The nonhomogeneous part of the ODE is a linear function, so we can start with assuming a particular solution of the form

Substituting these into the system gives




Put everything together to get a solution

that should match the solution in part (a).