1.用预报-校正法解下面常微分方程的初值问题。

2.用r-k方法求解初值问题：

import math as mh import matplotlib.pyplot as plt def g(x): ni=1/((x**2)+1)
return ni def no(jk1,h2):#question one with prediction correction method global
y def f(x,y): xv=-2*x*(y**2) return xv x1=jk1;x2=h2;h=0.01;y0=1
t=(x2-x1)/h;hi=int(t) for i in range(hi): n1=f(x1+i*h,y0); y1=y0+h*n1;
n2=f(x1+(i+1)*h,y1); y=y0+h/2*(n1+n2) y0=y return y x0=0#x的初值点 x1=2#想求的X值
r=1/5;y=0;no(x0,x1)
print('问题1:(保留6位小数)\n求解X={}时,y值\n预报校正法求解：{:.6};步长为0.01'.format(x1,y))
print('准确求解微分方程：{:.6}'.format(r))
print('预报和准确值的相对误差为：{:.6}%,绝对误差为:{:.6}'.format(abs(y-r)/r*100,abs(y-r)))
x1=[0];y1=[1];y2=[1] for i in range(300): y1.append(g(i*0.01))
x1.append(i*0.01) y2.append(no(x0,0.01+i*0.01))
plt.plot(x1,y1,c='r',label='Value') plt.plot(x1,y2,c='k',label='prediction')
plt.xlabel('X');plt.ylabel('Y');plt.title('P-M trace') plt.legend() plt.show()
def f(x,y):#question second with R-K.way xv=mh.sin(x)+mh.cos(y) return xv
c1=[];c2=[] for j in range(1000): x1=0;x2=10;h=0.01;y0=0;x0=0
t=(x2-x1)/h;hi=int(t)-j for i in range(hi): n1=f(x0,y0) n2=f(x0+h/2,y0+n1*h/2)
n3=f(x0+h/2,y0+n2*h/2) n4=f(x0+h,y0+n3*h) y=y0+h/6*(n1+2*n2+2*n3+n4) y0=y
x0=x0+h c2.append(y);c1.append(hi*h)
print('问题2:(保留6位小数)\n求解X={}时,y值\nR-K求解：{:.6};步长为0.01'.format(c1[900],c2[900]))
plt.plot(c1,c2,c='k') plt.xlabel('X');plt.ylabel('Y');plt.title('R-K trace')
plt.show()

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