Table 2.

Null forms S, integrating factors R (or the modified integrating factor Embedded Image) and first integrals I for the vector fields admitted by equation (3.1) using the Darboux polynomials given in equations (3.14)–(3.17).

vector fieldsnull forms (S)integrating factor (R)Embedded Imageintegrals
V1Embedded ImageR11Embedded ImageI11Embedded Image
R12Embedded ImageI12Embedded Image
R13Embedded ImageI13Embedded Image
R14Embedded ImageI14Embedded Image
V2Embedded ImageR21Embedded ImageEmbedded ImageEmbedded ImageI21Embedded Image
R22Embedded ImageI22I21
R23Embedded ImageEmbedded ImageEmbedded ImageI23I21
R24Embedded ImageEmbedded ImageEmbedded ImageI24I21
V3Embedded ImageR31Embedded ImageI31Embedded Image
R32Embedded ImageEmbedded ImageEmbedded ImageI32I31
R33Embedded ImageEmbedded ImageEmbedded ImageI33I31
R34Embedded ImageEmbedded ImageEmbedded ImageI34I31
V4Embedded ImageR41Embedded ImageEmbedded ImageEmbedded ImageI41I21
R42Embedded ImageEmbedded ImageEmbedded ImageI42I21
R43Embedded ImageEmbedded ImageEmbedded ImageI43I21
R44Embedded ImageI44I21
V5Embedded ImageR51Embedded ImageI51Embedded Image
R52Embedded ImageI52Embedded Image
R53Embedded ImageI53Embedded Image
R54Embedded ImageI54I13
V6Embedded ImageR61Embedded ImageEmbedded ImageEmbedded ImageI61Embedded Image
R62Embedded ImageEmbedded ImageEmbedded ImageI62Embedded Image
R63Embedded ImageEmbedded ImageEmbedded ImageI63Embedded Image
R64Embedded ImageEmbedded ImageEmbedded ImageI64I13
Embedded ImageEmbedded ImageEmbedded ImageEmbedded Image
V8Embedded ImageR81Embedded ImageEmbedded ImageEmbedded ImageI81Embedded Image
R82Embedded ImageI82Embedded Image
R83Embedded ImageEmbedded ImageEmbedded ImageI83Embedded Image
R84Embedded ImageEmbedded ImageEmbedded ImageI84Embedded Image
  • *In the vector field V7, we use the Darboux polynomials Embedded Image for calculating the integrating factor R.