they are asking you to solve
and
separately, then add the solutions together
the method of undetermined coefficients is the way to go here
see hereii) Solve the equation directly
take for example.
we must find the homogeneous solutions first. that is, solve
we assume a solution of
the characteristic equation is .........you should know how to get this.
or
Thus, the homogeneous solutions are: .
Now we find a particular solution. It must be of the form of the original right hand side, but not the same as the solution to the homogeneous. there is no danger of the latter case here.
so, assume is a particular solution.
and
plug these into the original differential equation, we get:
now simplify and equate coefficients to solve for C.
the solution to the equation will be of the form:
the others are solved in exactly the same way