## Solving third order differential equations with maximal symmetry group.(English)Zbl 0980.34001

After some articles concerning the algorithmic solving of second-order ODEs based on Lie methods [cf. ibid. 62, No. 1, 1-10 (1999; Zbl 0934.34001)], the author extends his work to the case of third-order ODEs with the maximal number of 7 symmetries.
Due to old results of K. Zorawski every such equation given in so-called actual variables $$x,y\equiv y(x)$$ has one of two possible forms, which may be recognized from corresponding conditions for some coefficients. From the other side such equation can be transformed to the canonical form $$v'''(u)= 0$$, where $$u$$ and $$v\equiv v(u)$$ are the so-called canonical variables given by $$u= \sigma(x,y)$$, $$v= \varrho(x,y)$$.
Using invariants, which are given in terms of $$\sigma$$, $$\varrho$$ and its derivatives and may be computed in terms of the coefficients cited above, the author constructs then for both cases systems of partial differential equations for the transformation functions $$\varrho(x,y)$$, $$\sigma(x,y)$$ leading from the actual to the canonical variables.
The crucial point is then to solve algorithmically some Riccati-like systems of quadratic PDEs or to get solutions by special methods as Loewy decomposition. There are instructive examples to illustrate the given method.

### MSC:

 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) 34A05 Explicit solutions, first integrals of ordinary differential equations

Zbl 0934.34001

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