Let y=pq where p and q are functions of x.
dy/dx=pq'+qp' and dy/dx=pq+e^x/x replacing y in the given DE by pq.
Equating these two expressions for dy/dx, q=q' (dq/dx=q; dq/q=dx; ln(q)=x; q=e^x)
qp'=e^xp'=e^x/x; dp/dx=1/x; dp=dx/x; p=ln(x).
y=pq=ke^xln(x) (k=constant of integration); but y=0 when x=e, so 0=ke^e and k=0.
CHECK: dy/dx=e^x/x+e^xln(x)=e^x/x+y; dy/dx-y=e^x/x