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This paper broadens the exponential utility function commonly used by risk-averse investors to the linear plus double exponential utility function, which is applicable in most cases. Thus it is of essential and supreme significance to conduct a research on its optimal investment portfolio in securities investment. This paper, by means of the non-difference curve method, carries out a research into the optimal portfolio decision-making by investors who have this type of utility function. The optimal decision-making and the ratio of optimal portfolio investment are derived. Finally, an actual case is given to verify the relevant results.

In the present paper, we study the formal first integrals of the well-known Lorenz system. First, for the simple condition of 0 = s we consider the form of the first integral. Furthermore, when 0 ≠ s and b is not a negative rational number, the Lorenz system dose not have any nontrivial formal first integral in a neighborhood of the origin.

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