Editing Linear differential equation (section)

==Higher order with variable coefficients==
A linear ordinary equation of order one with variable coefficients may be solved by [[quadrature (mathematics)|quadrature]], which means that the solutions may be expressed in terms of [[antiderivative|integrals]]. This is not the case for order at least two. This is the main result of [[Picard–Vessiot theory]] which was initiated by [[Émile Picard]] and [[Ernest Vessiot]], and whose recent developments are called [[differential Galois theory]].

The impossibility of solving by quadrature can be compared with the [[Abel–Ruffini theorem]], which states that an [[algebraic equation]] of degree at least five cannot, in general, be solved by radicals. This analogy extends to the proof methods and motivates the denomination of [[differential Galois theory]].

Similarly to the algebraic case, the theory allows deciding which equations may be solved by quadrature, and if possible solving them. However, for both theories, the necessary computations are extremely difficult, even with the most powerful computers.

Nevertheless, the case of order two with rational coefficients has been completely solved by [[Kovacic's algorithm]].

===Cauchy–Euler equation===
[[Cauchy–Euler equation]]s are examples of equations of any order, with variable coefficients, that can be solved explicitly. These are the equations of the form
<math display="block">x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \cdots + a_0 y(x) = 0,</math> 
where {{tmath|a_0, \ldots, a_{n-1} }} are constant coefficients.