Definition
Linear ordinary differential system
A system y prime equals A(t)y plus a prescribed forcing term.
A linear ODE system has the form , where and are prescribed. The system is homogeneous when and inhomogeneous when a nonzero source is present. Homogeneous here refers to the absence of a source, not to a scaling law.
Superposition
Linear combinations of solutions of the homogeneous system remain solutions. The difference of two solutions with the same source solves the homogeneous equation; consequently every inhomogeneous solution is a particular solution plus a homogeneous one. A fundamental matrix organizes this solution space.