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 functions.
Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.
The solutions of a homogeneous linear differential equation form a vector space.
In the ordinary case, this vector space has a finite dimension, equal to the order of the equation.
The basic differential operators include the derivative of order 0, which is the identity mapping.
In the case of an ordinary differential operator of order n, Carathéodory's existence theorem implies that, under very mild conditions, the kernel of L is a vector space of dimension n, and that the solutions of the equation Ly(x) = b(x) have the form
Typically, the hypotheses of Carathéodory's theorem are satisfied in an interval I, if the functions b, a0, ..., an are continuous in I, and there is a positive real number k such that |an(x)| > k for every x in I.
A homogeneous linear differential equation has constant coefficients if it has the form
It follows that the nth derivative of ecx is cnecx, and this allows solving homogeneous linear differential equations rather easily.
be a homogeneous linear differential equation with constant coefficients (that is a0, ..., an are real or complex numbers).
Factoring out eαx (which is never zero), shows that α must be a root of the characteristic polynomial
In the case where the characteristic polynomial has only simple roots, the preceding provides a complete basis of the solutions vector space.
In the case of multiple roots, more linearly independent solutions are needed for having a basis.
where k is a nonnegative integer, α is a root of the characteristic polynomial of multiplicity m, and k < m. For proving that these functions are solutions, one may remark that if α is a root of the characteristic polynomial of multiplicity m, the characteristic polynomial may be factored as P(t)(t − α)m. Thus, applying the differential operator of the equation is equivalent with applying first m times the operator
In the common case where the coefficients of the equation are real, it is generally more convenient to have a basis of the solutions consisting of real-valued functions.
Thus a real basis is obtained by using Euler's formula, and replacing
The best method depends on the nature of the function f that makes the equation non-homogeneous.
If, more generally, f is a linear combination of functions of the form xneax, xn cos(ax), and xn sin(ax), where n is a nonnegative integer, and a a constant (which need not be the same in each term), then the method of undetermined coefficients may be used.
Still more general, the annihilator method applies when f satisfies a homogeneous linear differential equation, typically, a holonomic function.
where (y1, ..., yn) is a basis of the vector space of the solutions and u1, ..., un are arbitrary constants.
The general form of a linear ordinary differential equation of order 1, after dividing out the coefficient of y′(x), is:
The solving method is similar to that of a single first order linear differential equations, but with complications stemming from noncommutativity of matrix multiplication.
Its solutions form a vector space of dimension n, and are therefore the columns of a square matrix of functions
A linear ordinary equation of order one with variable coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals.
This analogy extends to the proof methods and motivates the denomination of differential Galois theory.
Nevertheless, the case of order two with rational coefficients has been completely solved by Kovacic's algorithm.
There are efficient algorithms for both conversions, that is for computing the recurrence relation from the differential equation, and vice versa.
[3] It follows that, if one represents (in a computer) holonomic functions by their defining differential equations and initial conditions, most calculus operations can be done automatically on these functions, such as derivative, indefinite and definite integral, fast computation of Taylor series (thanks of the recurrence relation on its coefficients), evaluation to a high precision with certified bound of the approximation error, limits, localization of singularities, asymptotic behavior at infinity and near singularities, proof of identities, etc.