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Generalized Bilinear Differential Equations

We introduce a kind of bilinear differential equations by generalizing Hirota bilinear operators, and explore when the linear superposition principle can apply to the resulting generalized bilinear differential equations. Together with an algorithm using weights, two examples of generalized bilinear differential equations are computed to shed light on the presented general scheme for constructing bilinear differential equations which possess linear subspaces of solutions.