import numpy as np
from sympy import *
from IPython.display import *
import matplotlib.pyplot as plt
%matplotlib inline 
init_printing(use_latex=True)
var('a b t k C1')
u = Function("u")(t)
de = Eq(u+u.diff(t) * k)
display(de)
des = dsolve(de,u)
display(des)
des = des.subs(C1,0)
display(des)
f = Lambda((t,k,a,b),(a-b) * des.rhs + b)
display(Latex('$f(t,k,a,b) = ' + str(latex(f(t,k,a,b))) + '$'))
x = np.linspace(0,5,100)
plt.grid(True)
plt.xlabel('Time t seconds',fontsize=12)
plt.ylabel('$f(t,1,4,8)$',fontsize=16)
plt.plot(x,[f(t,1,4,8) for t in x],color='#008000')
plt.show()

$$k \frac{\partial}{\partial t} \operatorname{u}{\left (t \right )} + \operatorname{u}{\left (t \right )} = 0$$

$$\operatorname{u}{\left (t \right )} = e^{\frac{C_{1} - t}{k}}$$

$$\operatorname{u}{\left (t \right )} = e^{- \frac{t}{k}}$$

$f(t,k,a,b) = b + \left(a - b\right) e^{- \frac{t}{k}}$