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Special symmetries to standard Riccati equations and applications. (English) Zbl 1202.34007
The general Riccati equation $$\frac{d\phi(\xi)}{d\xi}=p(\xi)\phi^2(\xi)+q(\xi)\phi(\xi)+r(\xi) \tag1$$ is studied. Here, $p,q$ and $r$ are continuous functions, defined on some interval $[a,b]\subseteq\mathbb{R}$. Using Lie group symmetry, the authors obtain new integrability conditions for the generalized Riccati equation. Using this condition, 7 families of Riccati equations in standard form (1) are obtained which are integrable by quadratures. The obtained results are applied to construct travelling wave solutions for nonlinear evolution equations.

##### MSC:
 34A05 Methods of solution of ODE 34C14 Symmetries, invariants (ODE) 34A34 Nonlinear ODE and systems, general
##### Keywords:
Riccati equation; Lie groups; integrability condition
##### Software:
PDESpecialSolutions
Full Text:
##### References:
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