Question

格林公式sin(x+y)dx+(xcos(x+y))(dx+dy)=0求通解

Answer 1

解：∵sin(x+y)dx+(xcos(x+y))(dx+dy)=0
∴sin(x+y)dx+(xcos(x+y))d(x+y)=0..........(1)
在(1)中，令M=sin(x+y)，N=xcos(x+y)
∵αM/α(x+y)=cos(x+y) (αM/α(x+y)表示M关于(x+y)的偏导数，其它类同)
αN/αx=cos(x+y)
∴αM/α(x+y)=αN/αx
则由格林公式知，方程(1)是一个全微分方程
即由sin(x+y)dx+(xcos(x+y))d(x+y)=0
==>sin(x+y)dx+xd(sin(x+y))=0
==>d(x*sin(x+y))=0
==>∫d(x*sin(x+y))=0
==>x*sin(x+y)=C (C是积分常数)
故原方程的通解是x*sin(x+y)=C。