Group Classification of a Generalized Lane-Emden System

Table 3

Lie symmetries for 𝑛=1, for various functions 𝐻(𝑣) and 𝐺(𝑢).

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𝐻(𝑣) arbitrary, 𝐺(𝑢)=𝑐, 𝑐 a constant

𝑋₁=𝜕𝜕𝑢,𝑋₂𝜕=ln𝑥𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+4𝑣)𝜕𝑢

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢) arbitrary

𝑋₁=𝜕𝜕𝑣,𝑋₂𝜕=ln𝑥𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕ln𝑥+4𝑢)𝜕𝑣

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𝐻(𝑣)=𝑑,𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂𝜕=ln𝑥𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕ln𝑥+4𝑢)𝜕𝑣, 𝑋₄=𝛽𝑥²𝜕𝜕𝜕𝑣−4𝜕𝑢,

𝑋₅=(𝛽𝑥²ln𝑥−𝛽𝑥²)𝜕𝜕𝜕𝑣−4ln𝑥𝜕𝑢,

𝑋₆=(16𝑑𝑥²𝜕+64𝑢)𝜕𝑢+(16𝛼𝑥²+64𝑣−𝛽𝑑𝑥⁴)𝜕𝜕𝑣,

𝑋₇𝜕=16𝑥𝜕𝑥−(8𝛼𝑥²−𝛽𝑑𝑥⁴)𝜕𝜕𝑣−(16𝑑𝑥²𝜕+32𝑢)𝜕𝑢

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑢^−𝑝, 𝛼,𝑝, and 𝛽 constants (𝛽,𝑝≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂𝜕=ln𝑥𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕ln𝑥+4𝑢)𝜕𝑣,

𝑋₄𝜕=2𝑥𝜕𝜕𝑥+4𝑢𝜕𝑢−(4𝑣𝑝+𝛼𝑝𝑥²𝜕−4𝑣)𝜕𝑣

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑒^−𝑘𝑢, 𝛼,𝑘, and 𝛽 constants (𝛽,𝑘≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂𝜕=ln𝑥𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+4𝑢)𝜕𝑣,

𝑋₄𝜕=4𝜕𝑢−(𝛼𝑘𝑥²𝜕+4𝑘𝑣)𝜕𝑣

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𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽ln𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂𝜕=ln𝑥𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+4𝑢)𝜕𝑣,

𝑋₄𝜕=2𝑥𝜕𝜕𝑥+4𝑢𝜕𝑢+(4𝑣−𝛽𝑥²)𝜕𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎and𝑏constants(𝑏≠0), 𝐺(𝑢)=𝑐,𝑐 constant

𝑋₁=𝜕𝜕𝑢,𝑋₂𝜕=ln𝑥𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+4𝑣)𝜕𝑢,𝑋₄𝜕=4ln𝑥𝜕𝑣−(𝑏𝑥²ln𝑥−𝑏𝑥²)𝜕,𝜕𝑢

𝑋₅=𝑏𝑥²𝜕𝜕𝜕𝑢−4,𝜕𝑣

𝑋₆𝜕=32𝑥𝜕𝑥−16𝑐𝑥²𝜕𝜕𝑣+(𝑏𝑐𝑥⁴𝜕+64𝑢)𝜕𝑢,

𝑋₇𝜕=16𝑥𝜕𝑥−(8𝑎𝑥²−𝑏𝑐𝑥⁴)𝜕𝜕𝑢−(16𝑐𝑥²𝜕+32𝑣)𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and𝑏 constants(𝑏≠0), 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼and𝛽constants(𝛽≠0)

𝑋₁𝜕=𝐹(𝑥)𝜕𝑣,𝑋₂𝜕=𝑊(𝑥)𝜕𝑢,𝑋₃𝜕=𝑢𝜕𝜕𝑢+𝑣𝜕𝑣,

𝑋₄𝜕=𝑏𝑣𝜕𝜕𝑢+𝛽𝑢𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎and𝑏constants(𝑏≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝, 𝛽and𝑝constants(𝛽,𝑝≠0)

𝑋₁𝜕=(𝑏𝑥+𝑏𝑝𝑥)𝜕𝜕𝑥+4𝑏𝑢𝜕𝜕𝑢+(2𝑎+2𝑏𝑣−2𝑎𝑝−2𝑏𝑝𝑣)𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎and𝑏constants(𝑏≠0), 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢, 𝛽and𝑘constants(𝛽,𝑘≠0)

𝑋₁𝜕=𝑏𝑘𝑥𝜕𝜕𝑥+4𝑏𝜕𝜕𝑢−(2𝑎𝑘+2𝑏𝑘𝑣)𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑣^−𝑚, 𝑎,𝑚,and𝑏constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝑐, 𝑐aconstant

𝑋₁=𝜕𝜕𝑢,𝑋₂𝜕=ln𝑥𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+4𝑣),𝜕𝑢

𝑋₄𝜕=2𝑥𝜕𝑥+(4𝑢−𝑎𝑚𝑥²𝜕−4𝑚𝑢)𝜕𝜕𝑢+4𝑣𝜕𝑣

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𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚and𝑏constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼and𝛽constants(𝛽≠0)

𝑋₁𝜕=(𝛽𝑥+𝛽𝑚𝑥)𝜕𝜕𝑥+(2𝛼−2𝛼𝑚+2𝛽𝑢−2𝛽𝑚𝑢)𝜕𝜕𝑢+4𝛽𝑣𝜕𝑣

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𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚and𝑏constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝,𝑝and𝛽constants(𝛽,𝑝≠0)

𝑋₁𝜕=(𝑝𝑚𝑥−𝑥)𝜕𝜕𝑥+(2𝑚𝑢−2𝑢)𝜕𝜕𝑢+(2𝑝𝑣−2𝑣)𝜕𝑣

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𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚and𝑏constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢, 𝑘and𝛽constants(𝛽,𝑘≠0)

𝑋₁𝜕=𝑘𝑚𝑥𝜕𝜕𝑥+(2𝑚−2)𝜕𝜕𝑢+2𝑘𝑣𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏𝑒^−𝑚𝑣, 𝑎,𝑚,and𝑏constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝑐, 𝑐constant

𝑋₁=𝜕𝜕𝑢,𝑋₂𝜕=ln𝑥𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+4𝑣)𝜕𝑢,𝑋₄=(𝑎𝑚𝑥²𝜕+4𝑚𝑢)𝜕𝜕𝑢−4𝜕𝑣

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𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚and𝑏constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼and𝛽constants(𝛽≠0)

𝑋₁𝜕=𝛽𝑚𝑥𝜕𝜕𝑥−(2𝛼𝑚+2𝛽𝑚𝑢)𝜕𝜕𝑢+4𝛽𝜕𝑣

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𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚and𝑏constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝, 𝛽and𝑝constants(𝛽,𝑝≠0)

𝑋₁𝜕=𝑚𝑝𝑥𝜕𝜕𝑥+2𝑚𝑢𝜕𝜕𝑢+(2𝑝−2)𝜕𝑣

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𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚and𝑏constants(𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢, 𝛽and𝑘constants(𝛽,𝑘≠0)

𝑋₁𝜕=𝑚𝑘𝑥𝜕𝜕𝑥+2𝑚𝜕𝜕𝑢+2𝑘𝜕𝑣,𝑋₂𝜕=(𝑚𝑘𝑥ln𝑥−𝑘𝑚𝑥)𝜕𝜕𝑥+2𝑚ln𝑥𝜕𝜕𝑢+2𝑘ln𝑥𝜕𝑣

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𝐻(𝑣)=𝑎+𝑏ln𝑣,𝑎,𝑚,and𝑏constants(𝑏≠0), 𝐺(𝑢)=𝑐, 𝑐constant

𝑋₁=𝜕𝜕𝑢,𝑋₂𝜕=ln𝑥𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+4𝑣)𝜕𝑢,𝑋₄𝜕=2𝑥𝜕𝑥+(4𝑢−𝑏𝑥²)𝜕𝜕𝜕𝑢+4𝑣𝜕𝑣

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