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In this paper, we focus on a general class of Schr\"odinger equations thatare time-dependent and quadratic in X and P. We transform Schr\"odingerequations in this class, via a class of time-dependent mass equations, to aclass of solvable time-dependent oscillator equations. This transformationconsists of a unitary transformation and a change in the ``time'' variable. Wederive mathematical constraints forthe transformation and introduce twoexamples.Comment: LaTeX, 18 pages, new format, edite

Time-dependent Schrödinger equations having isomorphic symmetry algebras. I. Classes of interrelated equations