求解dx/(x+t)=dy/(-y+t)=dt

求解dx/(x+t)=dy/(-y+t)=dt
求解dx/(x+t)=dy/(-y+t)=dt

求解dx/(x+t)=dy/(-y+t)=dt
∵dx/(x+t)=dy/(-y+t)=dt
==>dx/(x+t)=dt,dy/(-y+t)=dt
==>dx-xdt=tdt,dy+ydt=tdt
==>e^(-t)dx-xe^(-t)dt=te^(-t)dt,e^tdy+ye^tdt=te^tdt
==>d(xe^(-t))=d(-(t+1)e^(-t)),d(ye^t)=d((t-1)e^t)
==>xe^(-t)=C1-(t+1)e^(-t),ye^t=C2+(t-1)e^t (C1,C2是积分常数)
==>x=C1e^t-t-1,y=C2e^(-t)+t-1
∴此微分方程的通解是x=C1e^t-t-1,y=C2e^(-t)+t-1 (C1,C2是积分常数).