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The objective or this research is to characterize the mathematical conceptions of future teachers in the resolution of ordinary differential equations in the context of argumentation. The collective argumentation concepts, the mathematical conceptions and ordinary differential equations constitute the research conceptual framework, reflected in the implementation of the enriched Toulmin model, a methodological tool that allows us to reconstruct the teachers' conceptions from the mathematical argumentation. The findings of the research indicate that the mathematical conceptions of future teachers are grounded on representation systems and on the ways to prove solutions for ordinary differential equations in the case of orthogonal trajectories. [ABSTRACT FROM AUTHOR]