Solutions for certain classes of Riccati differential equation
arXiv:0707.2837 · doi:10.1088/1751-8113/40/35/012
Abstract
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.
10 pages