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
,
:

Condense the left side as the derivative of a product:

Integrate both sides and multipy by
to solve for
:

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





b. Find the eigenvalues of
:


Let
and
be the corresponding eigenvectors.
For
, we have

which means we can pick
and
.
For
, 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).