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)+a_n-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).