import numpy as np
from scipy import integrate
import matplotlib.pyplot as plt
import sympy

def plot_direction_field(x, y_x, f_xy, x_lim=(-5, 5), y_lim=(-5, 5), ax=None):
f_np = sympy.lambdify((x, y_x), f_xy, 'numpy')
x_vec = np.linspace(x_lim[0], x_lim[1], 20)
y_vec = np.linspace(y_lim[0], y_lim[1], 20)

if ax is None:  
    \_, ax = plt.subplots(figsize=(4, 4))  
# dx相邻两值的距离  
dx = x\_vec\[1\] - x\_vec\[0\]  
dy = y\_vec\[1\] - y\_vec\[0\]

for m, xx in enumerate(x\_vec):  
    for n, yy in enumerate(y\_vec):  
        Dy = f\_np(xx, yy) \* dx  
        Dx = 0.8 \* dx \*\* 2 / np.sqrt(dx \*\* 2 + Dy \*\* 2)  
        Dy = 0.8 \* Dy \* dy / np.sqrt(dx \*\* 2 + Dy \*\* 2)  
        ax.plot(\[xx - Dx / 2, xx + Dx / 2\], \[yy - Dy / 2, yy + Dy / 2\], 'b', lw=0.5)

ax.axis('tight')  
# y对x的偏导=f\_xy  
ax.set\_title(r"$%s$" % (sympy.latex(sympy.Eq(y\_x.diff(x), f\_xy))), fontsize=18)

return ax

自变量

x = sympy.symbols('x')

因变量

y = sympy.Function('y')
f = x - y(x) ** 2

f_np = sympy.lambdify((y(x), x), f)
y0 = 1
xp = np.linspace(0, 5, 100)

func 初始值 x的范围

yp = integrate.odeint(f_np, y0, xp)
xn = np.linspace(0, -5, 100)
yn = integrate.odeint(f_np, y0, xn)

画场线图

fig, ax = plt.subplots(1, 1, figsize=(4, 4))
plot_direction_field(x, y(x), f, ax=ax)
ax.plot(xn, yn, 'b', lw=2)
ax.plot(xp, yp, 'r', lw=2)
plt.show()

from sympy import *
x = symbols("x")
y = symbols("y",cls=Function)
eq1 = diff(y(x),x,2)-5*diff(y(x),x)+6*y(x)
eq2 = diff(y(x),x,2)-5*diff(y(x),x)+6*y(x)-x*exp(2*x)
res = dsolve(eq1,y(x))
res2 = dsolve(eq2, y(x))
print(res)
print(res2)

Eq(y(x), (C1 + C2*exp(x))*exp(2*x))
Eq(y(x), (C1 + C2*exp(x) - x**2/2 - x)*exp(2*x))

from sympy import *
x = symbols("x")
y = symbols("y",cls=Function)

eq1 = diff(y(x),x,2)-5*diff(y(x),x)+6*y(x)
eq2 = diff(y(x),x,2)-5*diff(y(x),x)+6*y(x)-x*exp(2*x)

res1 = dsolve(eq1,y(x),ics={y(0):1,diff(y(x),x).subs(x,0):0})
res2 = dsolve(eq2,y(x),ics={y(0):1,y(2):0})
print(res1)
print(res2)

Eq(y(x), (3 - 2*exp(x))*exp(2*x))
Eq(y(x), (-x**2/2 - x + 3*exp(x)/(-1 + exp(2)) + (-4 + exp(2))/(-1 + exp(2)))*exp(2*x))