python 求常微分方程 sympy库pythont4ngw的博客-

问题1：

$f”(x)-2f'(x) + f(x) = sin(x)$

f′′(x)−2f′(x)+f(x)=sin(x)

程序，如下

from sympy import * f = symbols('f', cls=Function) x = symbols('x') eq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x)) print(dsolve(eq, f(x))) 

结果

Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2) 

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附：布置考试中两题
1.利用python的Sympy库求解微分方程的解
$y=f(x)$

y=f(x)，并尝试利用matplotlib绘制函数图像

$f'(x)+f(x)+f^2(x)=0,qquad f(0)=1$

f′(x)+f(x)+f2(x)=0,f(0)=1

程序，如下

from sympy import * f = symbols('f', cls=Function) x = symbols('x') eq = Eq(f(x).diff(x,1)+f(x)+f(x)**2, 0) print(dsolve(eq, f(x))) C1 = symbols('C1') eqr = -C1/(C1 - exp(x)) eqr1 = eqr.subs(x, 0) print(solveset(eqr1 - 1, C1)) eqr2 = eqr.subs(C1, 1/2) # 画图 import matplotlib.pyplot as plt import numpy as np x_1 = np.arange(-5, 5, 0.1) y_1 = [-0.5/(0.5 - exp(x)) for x in x_1] plt.plot(x_1, y_1) plt.axis([-6,6,-10,10]) plt.grid() plt.show() 

结果

Eq(f(x), -C1/(C1 - exp(x))) FiniteSet(1/2) 

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2.利用python的Sympy库求解微分方程的解
$y=y(x)$

y=y(x)，并尝试利用matplotlib绘制函数图像

$y'(x)=y(x),qquad y(0)=1$

y′(x)=y(x),y(0)=1

程序，如下

from sympy import * y = symbols('y', cls=Function) x = symbols('x') eq = Eq(y(x).diff(x,1), y(x)) print(dsolve(eq, y(x))) C1 = symbols('C1') eqr = C1*exp(x) eqr1 = eqr.subs(x, 0) print(solveset(eqr1 - 1, C1)) eqr2 = eqr.subs(C1, 1) # 画图 import matplotlib.pyplot as plt import numpy as np x_1 = np.arange(-5, 5, 0.01) y_1 = [exp(x) for x in x_1] plt.plot(x_1, y_1, color='orange') plt.grid() plt.show() 

结果

Eq(y(x), C1*exp(x)) FiniteSet(1)