DSolve[{t*Derivative[1][x][t] == -t + 2*x[t], t^3*Derivative[1][y][t] == t - x[t] + t^2*y[t], t^4*Derivative[1][z][t] == t - x[t] - t^2*y[t] + t^3*z[t]},{x[t], y[t], z[t]},t]