# Method of characteristics with coupled ODEs

I am having trouble following the derivation in this paper https://arxiv.org/abs/1810.07775 using the method of characteristics. By using the method of characteristics, they derive the following ODEs that are satisfied along the characteristic curve:

$$\frac{dr}{dt} = 2\gamma r - K(t)\\ \frac{dK}{dt} = w^2r(t)$$ The solution they give for $$K(t), r(t)$$ is a bit complicated, but does not exactly satisfy the ODEs above. I get different expressions for $$K(t),r(t)$$, which seem to satisfy the above equations but produce different characteristic curves than the paper.

Specifically, I would like to know the following:

1. What is the exact solution to the above equations?

2. If your exact solution is different than the one in the paper, why?

• please expand in a few lines to explain your problem, it is good that you gave a reference but some added verbiage would make it a better question. Commented Mar 31 at 20:32

Just a quick numerical test, to solve for $$r(t),K(t)$$ with Python (I assume $$\gamma = 1/2, \omega=1$$):

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

def coupled_ode(y, t):
r, K = y
drdt = r - K
dKdt = r
return [drdt, dKdt]

t = np.linspace(40, 60, 1000)
sol = odeint(coupled_ode, [1.0, 1.0], t)

plt.plot(t, sol[:, 0], label='r(t), numerical')
plt.plot(t, sol[:, 1], label='K(t), numerical')
plt.legend()
plt.show()


Shows :

So I think it could be fluctuations in the form as $$\propto~ e^{\lambda_1 t}- e^{\lambda_2 t}$$

• This code/graph was very helpful to check my analytical results; there are small differences between them which I assume are due to numerical error. The issue is that your numerical solution here seems different than the solution in their paper, and I am unsure why that is the case. Commented Mar 31 at 22:15
• I'm not quite sure if it's different than the original solution. I would rather say that it should be the same. Just maybe, solution is not unique. Commented Apr 1 at 5:40