Simultanous differential equations in Python

Question

I wanna solve a simultaneous differential equation and based on Lorenz equations:

def f(xyz, t, rho, sigma, beta):
    x, y, z = xyz
    return [sigma * (y - x),
            x * (rho - z) - y,
            x * y - beta * z]

I Wrote this:

def f(xyz, t, rho, sigma, beta):
    x, y, z = xyz
    return [sigma * y(t).diff(t) + sigma * x + beta * y -77,
            x + rho * y - 61]

So basically I have another differential of y in the first equation and I tried to take derivative but it says:

TypeError: 'numpy.float64' object is not callable"

Can you tell me How can I solve such problems and for second orders of those?

Answer 1

You have a linear system for the derivatives, call them xp, yp. Fortunately it is triangular, so you can solve it via back-substitution. So first solve for yp then insert that for xp.

def f(xy,t):
    x,y = xy
    yp = 61 - (4*x + y)
    xp = 77 - (2*yp + 2*x + 5*x)
    return xp,yp

In general you could employ a linear-system solver numpy.linalg.solve or a more general solver like scipy.optimize.fsolve to extract the derivatives from a system of implicit equations (as far as possible, DAE - systems of differential-algebraic equations can also have this form).