New renormalization procedure for perturbed ordinally differential equations with given initial condition and prescribed secular terms

For perturbed ordinally differential equations, a procedure of renormalization group method is proposed. To uniquely obtain renormalized solutions for given initial conditions, the procedure assumes that the extra integral constants yielded by perturbative expansion are determined as they cancel out all the prescribed secular terms. A sufficient condition for the renormalized solutions coinciding with exact solutions is also stated.