The Gerdjikov-Ivanov equation which appears，in the fields of quanta field theory，weak nonlinear dispersive water wave，nonlinear optics， etc.， has been discussed. Nonlinear mathematical physics equation with higher order nonlinear terms is educed in the discussion of Gerdjikov-Ivanov equation. The Liénard equation is chosen as subsidiary ordinary differential equation, with the help of which and according to homogeneous balance principle，the Gerdjikov-Ivanov equation has been solved，and the envelope solitary wave solutions and envelope sinusoidal wave solutions have been obtained.