In this work, we apply the 'disconjugacy theory' and Elias's spectrum theory to study the disconjugacy $ u^{(4)} + \beta u''-\alpha u = 0 $ with two parameters $ \alpha,\beta\in\mathbb{R} $ and the spectrum structure of the linear operator $ u^{(4)} + \beta u''-\alpha u $ coupled with the clamped beam conditions $ u(0) = u'(0) = u(1) = u'(1) = 0 $. As the application of our results, we obtain the global structure of nodal solutions of the corresponding nonlinear analogue based on the bifurcation theory.
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