Need help solving a second order non-linear ODE in python

Question

I don\'t really know where to start with this problem, as I haven\'t had much experience with this but it is required to solve this part of the project using a computer.

Answer 1

I don't think you can solve your problem as stated: your initial conditions, with x = 0 and x' > 0 imply that the solution will be positive for some values very close to the starting point. So there is no b for which the solution is never positive...

Leaving that aside, to solve a second order differential equation, you first need to rewrite it as a system of two first order differential equations. Defining y = x' we can rewrite your single equation as:

x' = y
y' = -b/m*y - k/m*x - a/m*x**3 - g

x[0] = 0, y[0] = 5

So your function should look something like this:

def fun(z, t, m, k, g, a, b):
    x, y = z
    return np.array([y, -(b*y + (k + a*x*x)*x) / m - g])

And you can solve and plot your equations doing:

m, k, g, a = 1220, 35600, 17.5, 450000
tmax, dt = 10, 0.01
t = np.linspace(0, tmax, num=np.round(tmax/dt)+1)
for b in xrange(1000, 10500, 500):
    print 'Solving for b = {}'.format(b)
    sol = odeint(fun, [0, 5], t, args=(m, k, g, a, b))[..., 0]
    plt.plot(t, sol, label='b = {}'.format(b))
plt.legend()