Solving a differential equations but an error appears saying "'Add' object is not callable". I am using Jupyter Notebook

Question

To make it clearer I want to plot the solutions of the 2nd order differential equation of the damping oscillation for a pendulum. Link to wiki about the equations used : https://en.wikipedia.org/wiki/Harmonic_oscillator

from sympy.interactive import printing
printing.init_printing(use_latex=True)
import numpy as np
import scipy as sp
from sympy import*
mport sympy as syp
`from scipy.integrate import odeint
import matplotlib.pyplot as plt

t=syp.Symbol('t')
x=syp.Function('x')(t)
m=2.0
k=5.0
a=0.5
z=a/(2.0*np.sqrt(m*k))
w=np.sqrt(k/m)
eq=x.diff(t,t)+2.0*z*w*x.diff(t)+w**2.0*x
dsolve(eq,t,0,ics={eq(1.0):0,eq(2.0):5})

Answer 1

You're not constructing the ics argument as intended:

In [6]: dsolve(eq, ics={x.subs(t, 1.0): 0, x.subs(t, 2.0): 5})                                                                                 
Out[6]: 
                                                                                                 -0.125⋅t
x(t) = (-0.0346285740992263⋅sin(1.57619002661481⋅t) - 6.42012708343871⋅cos(1.57619002661481⋅t))⋅ℯ 

The answer comes out nicer (subjectively) if you don't use floats. Also I find it more natural to keep the variable x as the function x rather than the applied function x(t) e.g.:

In [15]: x = Function('x')                                                                                                                     

In [16]: x                                                                                                                                     
Out[16]: x

In [17]: x(t)                                                                                                                                  
Out[17]: x(t)

In [18]: eq = x(t).diff(t, 2) + x(t).diff(t)/4 + 5*x(t)/2                                                                                      

In [19]: eq                                                                                                                                    
Out[19]: 
         d                   
         ──(x(t))     2      
5⋅x(t)   dt          d       
────── + ──────── + ───(x(t))
  2         4         2      
                    dt       

In [20]: dsolve(eq, x(t), ics={x(1): 0, x(2): 5})                                                                                              
Out[20]: 
       ⎛   1/4    ⎛√159⋅t⎞                     ⎞  -t 
       ⎜5⋅ℯ   ⋅sin⎜──────⎟                     ⎟  ───
       ⎜          ⎝  8   ⎠      1/4    ⎛√159⋅t⎞⎟   8 
x(t) = ⎜────────────────── - 5⋅ℯ   ⋅cos⎜──────⎟⎟⋅ℯ   
       ⎜       ⎛√159⎞                  ⎝  8   ⎠⎟     
       ⎜    tan⎜────⎟                          ⎟     
       ⎝       ⎝ 8  ⎠                          ⎠