如何使用 Python 内置函数 odeint 求解微分方程？
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How to solve differential equation using Python builtin function odeint?
提问by Physicist
I want to solve this differential equations with the given initial conditions:
我想用给定的初始条件求解这个微分方程：
(3x-1)y''-(3x+2)y'+(6x-8)y=0, y(0)=2, y'(0)=3
the ans should be y=2*exp(2*x)-x*exp(-x)
答案应该是 y=2*exp(2*x)-x*exp(-x)
here is my code:
这是我的代码：
def g(y,x):
y0 = y[0]
y1 = y[1]
y2 = (6*x-8)*y0/(3*x-1)+(3*x+2)*y1/(3*x-1)
return [y1,y2]
init = [2.0, 3.0]
x=np.linspace(-2,2,100)
sol=spi.odeint(g,init,x)
plt.plot(x,sol[:,0])
plt.show()
but what I get is different from the answer. what have I done wrong?
但我得到的与答案不同。我做错了什么？
采纳答案by xnx
There are several things wrong here. Firstly, your equation is apparently
这里有几件事是错误的。首先，你的方程显然是
(3x-1)y''-(3x+2)y'-(6x-8)y=0; y(0)=2, y'(0)=3
(3x-1)y''-(3x+2)y'-(6x-8)y=0；y(0)=2, y'(0)=3
(note the sign of the term in y). For this equation, your analytical solution and definition of y2are correct.
（注意 y 中术语的符号）。对于这个方程，你的解析解和定义y2是正确的。
Secondly, as the @Warren Weckesser says, you must pass 2 parameters as yto g: y[0](y), y[1](y') and return their derivatives, y' and y''.
其次，正如@Warren Weckesser 所说，您必须将 2 个参数传递y给g: y[0](y), y[1](y') 并返回它们的导数 y' 和 y''。
Thirdly, your initial conditions are given for x=0, but your x-grid to integrate on starts at -2. From the docs for odeint, this parameter, tin their call signature description:
第三，您的初始条件是为 x=0 给出的，但是您要积分的 x 网格从 -2 开始。从 docs for odeint，这个参数，t在他们的调用签名描述中：
odeint(func, y0, t, args=(),...):
odeint(func, y0, t, args=(),...)：
t : array A sequence of time points for which to solve for y. The initial value point should be the first element of this sequence.
t : array 求解 y 的时间点序列。初始值点应该是这个序列的第一个元素。
So you must integrate starting at 0 or provide initial conditions starting at -2.
因此，您必须从 0 开始积分或提供从 -2 开始的初始条件。
Finally, your range of integration covers a singularity at x=1/3. odeintmay have a bad time here (but apparently doesn't).
最后，您的积分范围涵盖 x=1/3 处的奇点。odeint可能在这里过得不好（但显然没有）。
Here's one approach that seems to work:
这是一种似乎有效的方法：
import numpy as np
import scipy as sp
from scipy.integrate import odeint
import matplotlib.pyplot as plt
def g(y, x):
y0 = y[0]
y1 = y[1]
y2 = ((3*x+2)*y1 + (6*x-8)*y0)/(3*x-1)
return y1, y2
# Initial conditions on y, y' at x=0
init = 2.0, 3.0
# First integrate from 0 to 2
x = np.linspace(0,2,100)
sol=odeint(g, init, x)
# Then integrate from 0 to -2
plt.plot(x, sol[:,0], color='b')
x = np.linspace(0,-2,100)
sol=odeint(g, init, x)
plt.plot(x, sol[:,0], color='b')
# The analytical answer in red dots
exact_x = np.linspace(-2,2,10)
exact_y = 2*np.exp(2*exact_x)-exact_x*np.exp(-exact_x)
plt.plot(exact_x,exact_y, 'o', color='r', label='exact')
plt.legend()
plt.show()
