Question

微分方程题目如图

题目如图

Answer 1

[1+2e^(x/y)] dx/dy +2e^(x/y) (1-x/y) =0

令x/y=u，x=yu，则dx/dy=u+ydu/dy

代入原方程得

(1+2e^u) (u+ydu/dy) +2e^u (1-u)=0

ydu/dy=-(u+2e^u)/(1+2e^u)

(1+2e^u)du/(u+2e^u)=-dy/y

d(u+2e^u)/(u+2e^u)=-dy/y

ln|u+2e^u|=-ln|y|+ln|c|

u+2^u =c/y

x/y +2e^(x/y)=c/y

x+2ye^(x/y)=c【通解】