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).