The formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Using the well-known fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the so-called Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OreMorphisms package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a system of partial differential equations. Finally, we show explicit computations on different examples.
