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arXiv 0707.2837math-phmath.MP
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一类Riccati微分方程的解

Solutions for certain classes of Riccati differential equation

Nasser Saad, Richard L Hall, Hakan Ciftci

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10 pages
发表信息：
J. Phys. A 40, 10903 - 10914 (2007)
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摘要：

本文给出了一类Riccati方程y '（x）-λ_0（x）y（x）\pm y^2（x）= \pm s_0（x）的解析闭式解，其中λ_0（x），s_0（x）是C^{ \infty }-函数.本文证明了若δ_n=λ_n s_{n-1} -λ_{n-1}s_n=0，其中λ_{n}= λ_{n-1}^ \prime + s_{n-1} +λ_0λ_{n-1}和s_{n} = s_{n-1} ^ \prime +s_0λ_{k-1}，n= 1，2，.，则Riccati方程有解y（x）= \mp s_{n-1}（x）/λ_{n-1}（x）.推广了广义Riccati方程y ′（x）+P（x）y（x）+Q（x）y^2（x）=R（x）.

英文摘要：

We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-λ_0(x)y(x)\pm y^2(x)=\pm s_0(x), where λ_0(x), s_0(x) are C^{\infty}-functions. We show that if δ_n=λ_n s_{n-1}-λ_{n-1}s_n=0, where λ_{n}= λ_{n-1}^\prime+s_{n-1}+λ_0λ_{n-1} and s_{n}=s_{n-1}^\prime+s_0λ_{k-1}, n=1,2,..., then The Riccati equation has a solution given by y(x)=\mp s_{n-1}(x)/λ_{n-1}(x). Extension to the generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also investigated.

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