I'm trying to solve a differential equation numerically, and am writing an equation that will give me an array of the solution to each time point.

import numpy as np
import matplotlib.pylab as pltpi=np.pi
sin=np.sin
cos=np.cos
sqrt=np.sqrt
alpha=pi/4 
g=9.80665
y0=0.0
theta0=0.0sina = sin(alpha)**2
second_term = g*sin(alpha)*cos(alpha)x0 = float(raw_input('What is the initial x in meters?'))
x_vel0 = float(raw_input('What is the initial velocity in the x direction in m/s?'))
y_vel0 = float(raw_input('what is the initial velocity in the y direction in m/s?'))
t_f = int(raw_input('What is the maximum time in seconds?'))r0 = x0
vtan = sqrt(x_vel0**2+y_vel0**2)
dt = 1000
n = range(0,t_f)
r_n = r0*(n*dt)
r_nm1 = r0((n-1)*dt)
F_r = ((vtan**2)/r_n)*sina-second_term
r_np1 = 2*r_n - r_nm1 + dt**2 * F_r
data = [r0]for time in n:data.append(float(r_np1))
print data

I'm not sure how to make the equation solve for r_np1 at each time in the range n. I'm still new to Python and would like some help understanding how to do something like this.

First issue is:

n = range(0,t_f) r_n = r0*(n*dt)

Here you define n as a list and try to multiply the list n with the integer dt. This will not work. Pure Python is NOT a vectorized language like NumPy or Matlab where you can do vector multiplication like this. You could make this line work with

n = np.arange(0,t_f) r_n = r0*(n*dt),

but you don't have to. Instead, you should move everything inside the for loop to do the calculation at each timestep. At the present point, you do the calculation once, then add the same only result t_f times to the data list.

Of course, you have to leave your initial conditions (which is a key part of ODE solving) OUTSIDE of the loop, because they only affect the first step of the solution, not all of them.

So:

# Initial conditions
r0 = x0
data = [r0]# Loop along timesteps
for n in range(t_f):# calculations performed at each timestepvtan = sqrt(x_vel0**2+y_vel0**2)dt = 1000r_n = r0*(n*dt)r_nm1 = r0*((n-1)*dt)F_r = ((vtan**2)/r_n)*sina-second_termr_np1 = 2*r_n - r_nm1 + dt**2 * F_r# append result to output listdata.append(float(r_np1))# do something with output list
print data
plt.plot(data)
plt.show()

I did not add any piece of code, only rearranged your lines. Notice that the part:

n = range(0,t_f)
for time in n:

Can be simplified to:

for time in range(0,t_f):

However, you use n as a time variable in the calculation (previously - and wrongly - defined as a list instead of a single number). Thus you can write:

for n in range(0,t_f):

Note 1: I do not know if this code is right mathematically, as I don't even know the equation you're solving. The code runs now and provides a result - you have to check if the result is good.

Note 2: Pure Python is not the best tool for this purpose. You should try some highly optimized built-ins of SciPy for ODE solving, as you have already got hints in the comments.