How do you define a second derivative in the Python GEKKO module?

Question

I want to solve a second order differential equation with GEKKO. In the documentation there is only an example that shows you how to solve a first order equation. I can't figure out how to write the second derivative of y to make it work.

This is the example from the documentation for a first order differential equation.

from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt

m = GEKKO()
m.time = np.linspace(0,20,100)
k = 10

y = m.Var(value=5.0)
t = m.Param(value=m.time)
m.Equation(k*y.dt()==-t*y)
m.options.IMODE = 4
m.solve(disp=False)

plt.plot(m.time,y.value)
plt.xlabel('time')
plt.ylabel('y')

plt.show()

Answer 1

The first derivative can be declared as dy = y.dt() and the second derivative as ddy = dy.dt()

import numpy as np
import matplotlib.pyplot as plt


m = GEKKO()
m.time = np.linspace(0,20,100)
k = 10

y = m.Var(value = 5.0)
t = m.Param(value=m.time)
dy = m.Var(value = 0.0)
ddy = m.Var(value = -5/10)


m.Equations([
    k*y.dt()==-t*y,
    dy == y.dt(),
    ddy == dy.dt()

])

m.options.IMODE = 4
m.solve(disp=False)


plt.figure()
plt.plot(m.time,y.value, label = 'y')
plt.xlabel('time')
plt.plot(m.time, dy.value, label  = 'dy')
plt.xlabel('time')
plt.plot(m.time, ddy.value, label  = 'ddy')
plt.xlabel('time')
plt.legend()
plt.show()

You can find more information here : https://apmonitor.com/wiki/index.php/Apps/2ndOrderDifferential