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Pfaffian differential equations - methods of solution part2
- 1. Part II
- 2. If we express this equation in the form 𝑿. 𝑑𝒓 = 𝟎, then 𝑿 = (𝑷 𝒙, 𝒚 , 𝑸 𝒙, 𝒚 , 𝑹 𝒛 ).
- 3. Then 𝐶𝑢𝑟𝑙 𝑿 = 𝒊 𝒋 𝒌 ! !" ! !# ! !$ 𝑃(𝑥, 𝑦) 𝑄(𝑥, 𝑦) 𝑅(𝑧) = 𝒊 𝜕𝑅 𝜕𝑦 − 𝜕𝑄 𝜕𝑧 + 𝒋 𝜕𝑃 𝜕𝑧 − 𝜕𝑅 𝜕𝑥 + 𝒌 𝜕𝑄 𝜕𝑥 − 𝜕𝑃 𝜕𝑦 = 𝒊 0 − 0 + 𝒋 0 − 0 + 𝒌 𝜕𝑄 𝜕𝑥 − 𝜕𝑃 𝜕𝑦 = 𝒌 𝜕𝑄 𝜕𝑥 − 𝜕𝑃 𝜕𝑦 = 0,0, !% !" − !& !#
- 4. Hence, if the equa9on is integrable, i.e., if We will have !% !" − !& !# . 𝑅 𝑧 = 0 ⇒ !% !" − !& !# = 0 ⇒ !% !" = !& !#
- 10. Or So, the solution is 2 log 𝑥 + log 𝑢 − log 1 + 𝑢 + 𝑣 = log 𝐶 i.e., log x' + log u − log (1 + u + v)=log C ⇒ log "!( )*(*+ = log 𝐶 ⇒ "!( )*(*+ = 𝐶 10