DSolve[{Derivative[2][x][t] == -E^(-x[t]) + a*E^(2*x[t]) + Cos[y[t]]^2/E^(2*x[t]), Derivative[2][y][t] == (Cos[y[t]]*Sin[y[t]])/E^(2*x[t]) - Sec[y[t]]^2*Tan[y[t]]},{x[t], y[t]},t]