Solutions for certain classes of Riccati differential equation

Details Solutions for certain classes of Riccati differential equation Solutions for certain classes of Riccati differential equation

2007-07-19

arxiv.org/abs/0707.2837v1

J. Phys. A 40, 10903 - 10914 (2007)

Physics

Mathematical Physics

10 pages

Scientific paper

10.1088/1751-8113/40/35/012

We derive some analytic closed-form solutions for a class of Riccati equation y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are C^{\infty}-functions. We show that if \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{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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