DSolve[{x1'[t]==((-1)/T1)x1[t]+(1/T1)x2[t],x2'[t]==((-1)/T2)x2[t],x1[0]==1,x2[0]==0},{x1[t],x2[t]},t]

DSolve[{x1'[t]==((-1)/T1)x1[t]+(1/T1)x2[t],x2'[t]==((-1)/T2)x2[t],x1[0]==0,x2[0]==1},{x1[t],x2[t]},t]//Simplify