Math 2.0

Ordinary differential equations Definition

Kovacic's algorithm

Builds on Liouvillian functionDifferential Galois theoryPicard–Vessiot extension

Definition

Let r∈ℂ(x)r \in \C(x) be a rational function and consider the equation y′′=r yy'' = r\, y over the differential field ℂ(x)\C(x). Let GG be its differential Galois group, the group of differential automorphisms of a Picard–Vessiot extension. Then GG is an algebraic subgroup of SL2(ℂ)\mathrm{SL}_2(\C), and exactly one of the following holds.

  1. GG is conjugate to a group of upper triangular matrices; then there is a solution y=e∫ωy = e^{\int \omega} with ω∈ℂ(x)\omega \in \C(x).
  2. Case 1 fails and GG is conjugate to a group of matrices that are diagonal or anti-diagonal. Then some solution is y=e∫ωy = e^{\int \omega} with ω\omega algebraic of degree 22 over ℂ(x)\C(x).
  3. Cases 1 and 2 fail and GG is finite. Then all solutions are algebraic, and some solution is y=e∫ωy = e^{\int \omega} with ω\omega algebraic of degree 44, 66 or 1212.
  4. G=SL2(ℂ)G = \mathrm{SL}_2(\C); then the equation has no nonzero Liouvillian solution.

Kovacic's algorithm tests cases 1, 2 and 3 in turn. In each case it builds finitely many candidate exponents from the expansions of rr at its poles and at infinity. Then it searches for ω\omega by solving for a polynomial of bounded degree. It returns a Liouvillian solution, or the answer that none exists.

Remarks

Every equation y′′+a(x) y′+b(x) y=0y'' + a(x)\, y' + b(x)\, y = 0 with a,b∈ℂ(x)a, b \in \C(x) can be brought to the reduced form: z=y e12∫az = y\, e^{\frac12 \int a} satisfies z′′=rzz'' = r z with r=14a2+12a′−br = \tfrac14 a^2 + \tfrac12 a' - b. Example: for r=1r = 1, the solution y=exy = e^{x} has ω=y′/y=1∈ℂ(x)\omega = y'/y = 1 \in \C(x), so the equation falls in case 1. Non-example: for the Airy equation, r=xr = x, the algorithm finds no candidate in cases 1 to 3, so G=SL2(ℂ)G = \mathrm{SL}_2(\C) and no nonzero solution is Liouvillian. In case 3 the group modulo ±1\pm 1 is the tetrahedral, octahedral or icosahedral group, and these correspond to the degrees 44, 66 and 1212. Jerald Kovacic published the algorithm in 1986.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Algebraically closed field, concept, no evidence levelAlgebraically closed …conceptDerivation, concept, no evidence levelDerivationconceptDifferential Galois theory, concept, no evidence levelDifferential Galois t…conceptDifferential field, concept, no evidence levelDifferential fieldconceptField extension, concept, no evidence levelField extensionconceptGroup, concept, no evidence levelGroupconceptLinear differential equation of order n, concept, no evidence levelLinear differential e…conceptPicard–Vessiot extension, concept, no evidence levelPicard–Vessiot extens…conceptSplitting field, concept, no evidence levelSplitting fieldconceptVector space, definition, no evidence levelVector spacedefinitionPicard–Vessiot extension, definition, no evidence levelPicard–Vessiot extens…definitionRing, definition, no evidence levelRingdefinitionLinear map, definition, no evidence levelLinear mapdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDerivation, definition, no evidence levelDerivationdefinitionField, definition, no evidence levelFielddefinitionSubfield, definition, no evidence levelSubfielddefinitionDifferential field, definition, no evidence levelDifferential fielddefinitionField extension, definition, no evidence levelField extensiondefinitionLiouvillian function, definition, no evidence levelLiouvillian functiondefinitionDifferential Galois theory, definition, no evidence levelDifferential Galois t…definitionKovacic's algorithm, definition, no evidence levelKovacic's algorithmdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraically closed field concept
  2. Derivation concept
  3. Differential Galois theory concept
  4. Differential field concept
  5. Field extension concept
  6. Group concept
  7. Linear differential equation of order n concept
  8. Picard–Vessiot extension concept
  9. Splitting field concept
  10. Vector space definition
  11. Picard–Vessiot extension definition
  12. Ring definition
  13. Linear map definition
  14. Commutative ring definition
  15. Subring definition
  16. Unit of a ring definition
  17. Derivation definition
  18. Field definition
  19. Subfield definition
  20. Differential field definition
  21. Field extension definition
  22. Liouvillian function definition
  23. Differential Galois theory definition
  24. Kovacic's algorithm definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Picard%E2%80%93Vessiot_theory
About this page
TagMMEX2JZWGR
URIm2://MMEX2JZWGR
Kinddefinition
WritingW0 stub
MSC34M15, 12H05, 68W30
DefinesKovacic's algorithm
UsesLiouvillian function, Differential Galois theory, Picard–Vessiot extension
Changed2 November 2026

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