At first I thought the "t" was a distraction and wrote out the whole problem in terms of x and y only. It was easy to see that the result was an "exact differential" just as Peritus said. However, then I could not see any way to integrate that exact differential!
But if t= x/y, the x= yt and dx= ydt+ tdy. In terms of t and y only the differential becomes [tex](1+ e^t)(ydt+ tdy)+ (1-t)e^t dy= (t+ e^t)dt+ y(1+ e^t)dy and that's easy to integrate!