Can you write a python script to solve this differential equation:
f" + f" + f = 0 with f(0) = 0
user
Sure, I can help you with that. This is a standard second-order linear homogeneous differential equation with constant coefficients. The general solution for such a differential equation is of the form:

f(t) = A*cos(ωt) + B*sin(ωt)

where ω = √(c_2), A and B are arbitrary constants of integration. In your case, the coefficients are C = -1, D = -2, which gives:

ω = √(-2) = -1, -1, 1 or 1

Since we want the solution to be real, we ignore the two roots of -1. So, the general solution is:

f(t) = A*cos(t) + B*sin(t)

where A and B are arbitrary constants of integration.

To find these constants, we plug in the initial condition f(0) = 0 and get:

0 = A*cos(0) + B*sin(0)

0 = A

Plugging into the general solution, we get:

f(t) = B*sin(t)

So, the particular solution is:

f(t) = B*sin(t)

where B is an arbitrary constant. You can use this formula to find the solution for your specific initial condition.
assistant
Can you write a python script to solve this differential equation:
f" + f" + f = 0 with f(0) = 0
user
Sure, I can help you write a Python script to solve this differential equation using the `scipy.integrate.solve_ivp` function. This is a second order differential equation, so we'll need to convert it into a system of first order differential equations.

Here's a Python script that does this:

```python
import numpy as np
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt

# Define the system of first order ODEs
def system(t, y):
    f, g = y
    dfdt = g
    dgdt = -f - g
    return dfdt, dgdt

# Initial conditions
t0 = 0
y0 = [0, 1]  # f(0) = 0, f'(0) = 1

# Solve the ODE system on the interval [0, 10]
t_eval = np.linspace(0, 10, 1000)
sol = solve_ivp(system, [t0, 10], y0, t_eval=t_eval)

# Extract the solutions for f and f'
f = sol.y[0]
fp = sol.y[1]

# Plot the solutions
fig, ax = plt.subplots(2, sharex=True, figsize=(8, 6))
ax[0].plot(sol.t, f)
ax[0].set_ylabel('f')
ax[1].plot(sol.t, fp)
ax[1].set_ylabel("f'")
ax[1].set_xlabel('t')
plt.show()
```

This script first defines the system of first order ODEs, then sets the initial conditions and the time interval on which to solve the ODE. It then calls `solve_ivp` to solve the ODE system, and finally plots the solutions for `f` and `f'`.

Please note that the initial condition for `f'(0)` is set to 1 because if it were 0, the solution would be trivial (`f(t) = 0` for all `t`). If you have a specific non-zero value for `f'(0)`, you can replace 1 with that value in the `y0` array.

Also, please make sure to have the necessary Python libraries installed in your environment. You can install them using pip:

```bash
pip install numpy scipy matplotlib
```
assistant