Editing Linear differential equation (section)

==Types of solution==
{{anchor|solving by quadrature}}
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be '''solved by quadrature''', which means that the solutions may be expressed in terms of [[antiderivative|integrals]]. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, [[Kovacic's algorithm]] allows deciding whether there are solutions in terms of integrals, and computing them if any.

The solutions of homogeneous linear differential equations with [[polynomial]] coefficients are called [[holonomic function]]s. This class of functions is stable under sums, products, [[derivative|differentiation]], [[antiderivative|integration]], and contains many usual functions and [[special function]]s such as [[exponential function]], [[logarithm]], [[sine]], [[cosine]], [[inverse trigonometric functions]], [[error function]], [[Bessel function]]s and [[hypergeometric function]]s. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of [[calculus]], such as computation of [[antiderivative]]s, [[limit (mathematics)|limits]], [[asymptotic expansion]], and numerical evaluation to any precision, with a certified error bound.