An elementary proof by contradiction of the exactness condition for first-order ordinary differential equations

• J. J. Bissell ^[1]
  1. [1] University of York

     University of York

     Reino Unido

     
• Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 52, Nº. 6, 2021, págs. 965-971
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • The ability to distinguish between exact and inexact differentials is an important part of solving first-order differential equations of the form Adx + Bdy = 0, where A(x, y) = 0 and B(x, y) = 0 are functions of x and y However, although most undergraduate textbooks motivate the necessary condition for exactness, i.e. the requirement that ∂A/∂y = ∂B/∂x, simple proofs that this condition is sufficient are not always given. Here we demonstrate that the condition is sufficient using a novel proof by contradiction. Our approach is compared to other proofs existing in the literature, and considered in the context of personalized learning.