Differential equations

  • Elementary Theory
  • Philip Spain


Let I be an interval contained in R, not reducing to a single point, E atopological vector space over R, and A and B two open subsets of E. Let (x, y, t) ↦ g(x, y, t) be a continuous map of A × B × I into E; to every differentiable map u of I into A whose derivative takes its values in B we associate the map tg(u(t), u’(t), t) of I into E, and denote it by \( \tilde g\)(u); so \( \tilde g\)is defined on the set D(A, B) of differentiable functions of I into B whose derivatives have their values in B. We shall say that the equation \( \tilde g\)(u) = 0 is a differential equation in u (relative to the real variable t); a solution of this equation is also called an integral of the differential equation (on the interval I); it is a differentiable map of I into A, whose derivative takes values in B, such that g(u(t), u’(t), t) = 0 for every t ∈ I. By abuse of language we shall write the differential equation \(\tilde g\)(u) = 0 in the form
$$g\left( {x,x',t} \right) = 0, $$
on the understanding that x belongs to the set D(A, B).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (I).
    Jakob Bernoulli, Opera, 2 vol., Genève (Cramer-Philibert), 1744.Google Scholar
  2. (II).
    L. Euler, Opera Omnia: Institutiones calculi integralis, (1) t. XII, Leipzig-Berlin (Teubner), 1914.zbMATHGoogle Scholar
  3. (III).
    J.-L. Lagrange, Œuvres, Paris (Gauthier-Villars), 1867–1890: a) Solution de différents problèmes de calcul intégral, t. I, p. 471;Google Scholar
  4. (III).
    J.-L. Lagrange, Œuvres, Paris (Gauthier-Villars), 1867–1890: b) Sur le mouvement des nœuds des orbites planétaires, t. IV, p. 111.Google Scholar
  5. (IV).
    A.-L.Cauchy, Œuvrescomplètes,(2),t.XI,Paris(Gauthier-Villars),1913, p. 399 (= Exercices d’Analyse, Paris, 1840, t. I, p. 327).Google Scholar
  6. (IV bis).
    A.-L. Cauchy, dans Leçons de calcul différentiel et de calcul intégral, rédigées principalement d’après les méthodes de M. A.-L. Cauchy, par l’abbé Moigno, t. II, Paris, 1844.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Elementary Theory
  • Philip Spain
    • 1
  1. 1.Department of MathematicsUniversity of GlasgowGlasgowScotland

Personalised recommendations