Abstract—Special Lie group of transformations are considered. Multi-parameter transformations of translational, scaling, spiral and combined types are employed. Both ordinary and partial differential equations are considered. The general forms of the differential equations admitting such special transformations are given by theorems. The theorems are proven and sample differential equations are discussed as direct applications of the theorems. In this paper, we conduct the first systematic classification of multi-parameter special symmetries for ODEs and PDEs, and derive the general forms of differential equations admitting multi-parameter translational, scaling, spiral, and combined transformations, which fill the gap that the existing single-parameter symmetry classification cannot cover complex multi-parameter physical scenarios.

Keywords—differential equations, special lie group of transformations

Cite: Mehmet Pakdemirli, "Determining the Structures of the Differential Equations Admitting Multi-parameter Special Symmetries," International Journal of Applied Physics and Mathematics, vol. 16, no. 2, pp. 26-35, 2026.

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