Table 1.

Null forms (S), integrating factors (R), characteristics (Q), vector fields (V), Darboux polynomials (F) and first integrals (I) admitted by equation (3.1).

SRaQ=XVFI
Embedded ImageEmbedded ImageEmbedded ImageV1Embedded ImageEmbedded Image
Embedded ImageEmbedded ImageV5Embedded ImageEmbedded Image
Embedded ImageEmbedded ImageV6Embedded ImageI1
Embedded ImageEmbedded ImageEmbedded ImageV4Embedded ImageEmbedded Image
Embedded ImageEmbedded ImageV2Embedded ImageI2
Embedded ImageEmbedded ImageV8Embedded ImageEmbedded Image
Embedded ImageEmbedded ImageEmbedded ImageV3Embedded ImageEmbedded Image
Embedded ImageEmbedded ImageEmbedded ImageV7Embedded ImageEmbedded Image
  • aOne may note that when the pair (S,R) gives the null form and integrating factor for the first integral I, then the pair Embedded Image is also the null form and integrating factor for the first integral f(I) as shown in Chandrasekar et al.[5,26]. This fact is used to find the required R in this table.