A complementary solution is the solution to a component of the general solution that assumes that the differential equation is homogeneous.
The other component of the general solution is the particular solution.
review and revise below #reorganize
While the definition can sometimes change depending on context, the complementary solution as used within this database is defined to be the solution to the homogeneous equation. That is that if you are given a nonhomogeneous differential equation you simply set it equal to zero and solve for the roots.
For example, if given the differential equation
which when evaluated using characteristic equation comes out to be:
The complimentary solution is generally designated by using a subscript "c."
See Undetermined Coefficients - Paul's Online Notes and Stack Exchange for source and more information.