Quaternion-valued differential equations (QDEs) are a new kind of differential equations that have many applications in quantum mechanics, fluid mechanics, Frenet frame in differential geometry, kinematic modeling, attitude dynamics, Kalman filter design, spatial rigid body dynamics, etc. The largest difference between QDEs and ordinary differential equations (ODEs) lies in the algebraic structure. Due to the non-commutativity of the quaternion algebra, the set of all the solutions to the linear homogenous QDEs is completely different from ODEs. It is actually a right free module, not a linear vector space.