ABSTRACT
Linear higher-order constant-coefficient differential equations are capable of modelling many fundamental physical processes, so they represent an important and useful class of differential equations. The complementary function is the solution of the homogeneous form of the differential equation, and so contains all of the arbitrary constants that are always present in a general solution. The complementary function is shown to be determined as a result of finding the roots of a polynomial derived from the homogeneous form of the differential equation. The values of these constants are found by substituting the functional form into the differential equation and equating coefficients of corresponding terms on each side of the equation to make the result an identity. Boundary value problems are more complicated and, depending on the differential equation involved and the associated boundary conditions, it is shown that they may have a unique solution, a non-unique solution or no solution at all.
