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**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.

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.

Reviewer: G.Czichowski (Greifswald)

### 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 |