Exercise_04:Chapter 1 problem 1.5:The decay of two kinds of particles

一.摘要：

The corresponding rate equations are:

$$\dot{N_A}=\frac{N_B-N_A}{\tau},①$$
$$\dot{N_B}=\frac{N_A-N_B}{\tau}.②$$

solve this system of equations for the numbers of
nuclei,$N_A$and$N_B$,as functions of time.Consider different initial conditions,such as $N_A=100,N_B=0$,etc,and take $*t*=1s$.Show your numerical results.

二.背景介绍（公式推导）：

①+②，得 $$\dot{N_A}+\dot{N_B}=0,$$

故 $$N_A+N_B=constant,③$$

③代入①，得 $$\dot{N_A}=\frac{c-2N_A}{\tau},$$

因此， $$N_A(t)=0.5*c-0.5*(c-2N_A(0))*e^(\frac{-2t}{\tau}).$$

$$N_B(t)=0.5*c+0.5*(c-2N_A(0))*e^(\frac{-2t}{\tau}).$$

三.正文：

1.Results of The Decay Problem:

2.Testing The Program

(1).with any exact results:

(2).with different "step sizes":

##①.由Testing The Program可知，图像一是符合正确结果的（可靠的），并且具有稳定性；
##②.如图一所示，当$N_A=N_B=50$,系统达到稳定状态，此时$\dot{N_A}=\dot{N_B}=0$.

在此输入正文