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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 is given as an introduction to the types of equations that can be solved by DSolve.
Name of Equation
General Form
Date of Discovery
Separabley (x)f (x)g (y)1691G. Leibniz
Homogeneous1691G. Leibniz
Linear first-order ODEy (x)+P (x)y (x)Q (x)1694G. Leibniz
Bernoulliy (x)+P (x)y (x)Q (x)1695James Bernoulli
Riccatiy (x)f (x)+g (x)y (x)+h (x)y (x)21724Count Riccati
Exact first-order ODEMdx+Ndy0 with 1734L. Euler
Clairauty (x)xy (x)+f (y (x))1734A-C. Clairaut
Linear with constant coefficientsy (n) (x)+an-1y (n-1) (x)+...+a0y (x)P (x) with a1 constant1743L. Euler
Hypergeometricx (1-x)y (x)+ (c- (a+b+1)x)y (x)-aby (x)01769L. Euler
Legendre (1-x2)y (x)-2xy (x)+n (n+1)y (x)=01785M. Legendre
Besselx2y (x)+xy (x)+ (x2-n2)y (x)=01824F. Bessel
Mathieuy (x)+ (a-2qcos (2x))y (x)=01868E. Mathieu
Abely (x)f (x)+g (x)y (x)+h (x)y (x)2+k (x)y (x)31834N. H. Abel
Chiniy (x)f (x)+g (x)y (x)+h (x)y (x)n1924M. 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).