Solving homogeneous systems of ODEs with constant coefficients and of arbitrary order is a straightforward matter. They are solved by converting them to a system of first-order ODEs.
In general, systems of linear ODEs with non-constant coefficients can only be solved in cases where the coefficient matrix has a simple structure, as illustrated in the following examples.
As for single ODEs, there are sophisticated modern algorithms for solving systems of ODEs with rational coefficients.
The systems considered so far have all been homogeneous. If the system is inhomogeneous (that is, if there are terms free from any dependent variables and their derivatives), DSolve
applies either the method of variation of parameters
or the method of undetermined coefficients
to find the general solution.
Particular solutions to the system can be obtained by assigning values to the constants C