Abstract
In this paper, from the expanded class of the second-order linear differential equations, a subclass will be obtained for which a new condition for reductability according to Frobenius will be obtained. Explicit formulas of a particular solution for the subclass of the second-order linear differential equations are obtained. The third-order shortened Lorenz system was obtained from the Modified Lorenz system. In this paper, a particular solution of the third-order shortened Lorenz system via the integrability of the subclass of the second-order linear differential equations under certain conditions is obtained. At the end, via graphical visualization in Mathematica, the theory is supported by examples.
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