This paper presents a novel approach for the numerical solution of differential linear matrix inequalities. The solutions are searched in the class of piecewise-quadratic functions with symmetric matrix coefficients to be determined. To limit the numbers of unknowns, congruence constraints are considered to guarantee continuity of the solution and of its derivative. In Example section, some control problems involving differential linear matrix inequalities are considered and solved in order to compare the proposed approach with alternative approximation methods adopted in the literature.