用分离变量法求通解dx+xydy=y^2dx+ydy

Question

用分离变量法求通解dx+xydy=y^2dx+ydy

Answer 1

∵dx+xydy=y^2dx+ydy
==>y(x-1)dy=(y^2-1)dx
==>2ydy/(y^2-1)=2dx/(x-1)
==>d(y^2-1)/(y^2-1)=2d(x-1)/(x-1)
==>∫d(y^2-1)/(y^2-1)=2∫d(x-1)/(x-1) (积分)
==>ln│y^2-1│=2ln│x-1│+ln│C│ (C是任意常数)
==>y^2-1=C(x-1)^2
==>y^2=1+C(x-1)^2
∴此方程的通解是y^2=1+C(x-1)^2。