Overview of Ordinary Differential Equations (ODEs)

There are four major areas in the study of ordinary differential equations that are of interest in pure and applied science.

  • Exact solutions, which are closed-form or implicit analytical expressions that satisfy the given problem.
  • Numerical solutions, which are available for a wider class of problems, but are typically only valid over a limited range of the independent variables.
  • Qualitative theory, which is concerned with the global properties of solutions and is particularly important in the modern approach to dynamical systems.
    • Existence and uniqueness theorems, which guarantee that there are solutions with certain desirable properties provided a set of conditions is fulfilled by the differential equation.

    Of these four areas, the study of exact solutions has the longest history, dating back to the period just after the discovery of calculus by Sir Isaac Newton and Gottfried Wilhelm von Leibniz. The following table introduces the types of equations that can be solved by DSolve.

    name of equation
    general form
    date of discovery
    mathematician
    separable1691G. Leibniz
    homogeneous1691G. Leibniz
    linear first-order ODE1694G. Leibniz
    Bernoulli1695James Bernoulli
    Riccati1724Count Riccati
    exact first-order ODE with 1734L. Euler
    Clairaut1734A-C. Clairaut
    linear with constant coefficients with constant1743L. Euler
    hypergeometric1769L. Euler
    Legendre1785M. Legendre
    Bessel1824F. Bessel
    Mathieu1868E. Mathieu
    Abel1834N. H. Abel
    Chini1924M. Chini

    Examples of ODEs belonging to each of these types are given in other tutorials (clicking a link in the table will bring up the relevant examples).