New Nonlinear Systems Admitting Virasoro-Type Symmetry Algebra and Group-Invariant Solutions

Table 2

Solutions to system (21) of Case 5.

𝑟𝑖	𝑔(𝑡)	𝑝(𝑥,𝑦,𝑡)	𝑟(𝑥,𝑦,𝑡)
𝑟3	1	1√𝑦+2𝐶1+12𝑥	ℎ(𝑡)√𝑦
𝑟6	1	±𝑥ln𝑦−2𝑥ln𝑥+𝐶1𝑥	𝑘𝑦,𝑡𝑥
𝑟6	1	𝑥ln𝑦+2𝑒^∫−ℎ(𝑡)d𝑡∫^𝑡𝑒^ℎ(𝑎)𝑥/∫−ℎ(𝑡)d𝑡d𝑎+𝑒^−∫ℎ(𝑡)d𝑡𝐹𝑒^𝑥/𝑒∫−ℎ(𝑡)d𝑡	ℎ(𝑡)𝑥
𝑟7	1	−𝑥ln𝑦−2𝑒^∫−ℎ(𝑡)d𝑡∫^𝑡𝑒^ℎ(𝑎)𝑥/∫−ℎ(𝑡)d𝑡d𝑎+𝑒^−∫ℎ(𝑡)d𝑡𝐹𝑒^𝑥/𝑒∫−ℎ(𝑡)d𝑡	ℎ(𝑡)𝑥
𝑟13	1	𝑦^1/𝛼ℎ(𝑡)	𝑥ℎ̇(𝑡)(2/𝛼−1)ℎ(𝑡)+𝑘𝑦,𝑡
𝑟3	𝑡	𝑥ln𝑥𝑡+𝐶1𝑥+𝐶2𝑦^1/2+𝑥ℎ(𝑡)	−𝑥ln𝑥+𝑥𝑡+𝑥𝑡ℎ̇(𝑡)+𝐹(𝑡)√𝑦