This is documentation for Mathematica 6, which was
based on an earlier version of the Wolfram Language.
 Mathematica Tutorial

# 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 Mathematician Separable y (x)f (x)g (y) 1691 G. Leibniz Homogeneous 1691 G. Leibniz Linear first-order ODE y (x)+P (x)y (x)Q (x) 1694 G. Leibniz Bernoulli y (x)+P (x)y (x)Q (x) 1695 James Bernoulli Riccati y (x)f (x)+g (x)y (x)+h (x)y (x)2 1724 Count Riccati Exact first-order ODE Mdx+Ndy0 with 1734 L. Euler Clairaut y (x)xy (x)+f (y (x)) 1734 A-C. Clairaut Linear with constant coefficients y (n) (x)+an-1y (n-1) (x)+...+a0y (x)P (x) with a1 constant 1743 L. Euler Hypergeometric x (1-x)y (x)+ (c- (a+b+1)x)y (x)-aby (x)0 1769 L. Euler Legendre (1-x2)y (x)-2xy (x)+n (n+1)y (x)=0 1785 M. Legendre Bessel x2y (x)+xy (x)+ (x2-n2)y (x)=0 1824 F. Bessel Mathieu y (x)+ (a-2qcos (2x))y (x)=0 1868 E. Mathieu Abel y (x)f (x)+g (x)y (x)+h (x)y (x)2+k (x)y (x)3 1834 N. H. Abel Chini y (x)f (x)+g (x)y (x)+h (x)y (x)n 1924 M. 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).