Symmetries and ̄ rst integrals of ordinary di ®
نویسنده
چکیده
This paper describes a new symmetry-based approach to solving a given ordinary di¬erence equation. By studying the local structure of the set of solutions, we derive a systematic method for determining one-parameter Lie groups of symmetries in closed form. Such groups can be used to achieve successive reductions of order. If there are enough symmetries, the di¬erence equation can be completely solved. Several examples are used to illustrate the technique for transitive and intransitive symmetry groups. It is also shown that every linear second-order ordinary di¬erence equation has a Lie algebra of symmetry generators that is isomorphic to sl(3). The paper concludes with a systematic method for constructing rst integrals directly, which can be used even if no symmetries are known.
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