## Solving second order ordinary differential equations with maximal symmetry group.(English)Zbl 0934.34001

The author investigates the second-order ordinary differential equation $$y''+{A y'}^3+ {B y'}^2+C y' + D =0$$ with $$A,B,C,D\in Q(x,y)$$. By a nonlinear change of coordinates this is equivalent to a simpler equation. The functions involved in the transformation satisfy a system of linear partial differential equations. The proof is based on Lie symmetry and Janet bases. This algorithmic approach is illustrated by several examples. In the appendix a short introduction into Loewy decomposition for the solution of partial differential equations is presented.

### MSC:

 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 68W30 Symbolic computation and algebraic computation 34A05 Explicit solutions, first integrals of ordinary differential equations 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)

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