How can I solve a simultaneous equation in Python using sympy

Question

I have the two following equations:

40 = Vmax*5.041667 + (Vmax**2/Amax)*sympy.exp((-Amax/Vmax)*5.041667) - (Vmax**2/Amax)
20 = Vmax*2.897628 + (Vmax**2/Amax)*sympy.exp((-Amax/Vmax)*2.897628) - (Vmax**2/Amax)   

I wrote the following codes in sympy to solve this simultaneous equation to solve for Vmax and Amax

import sympy as sp
Vmax, Amax = symbols('Vmax, Amax')

eq1 = Vmax*5.041667 + (Vmax**2/Amax)*sympy.exp((-Amax/Vmax)*5.041667) - (Vmax**2/Amax)
eq2 = Vmax*2.897628 + (Vmax**2/Amax)*sympy.exp((-Amax/Vmax)*2.897628) - (Vmax**2/Amax)         

print(nsolve((eq1, eq2), (Vmax, Amax), (40,20)))

This code gives the wrong answer. And the following code which I think is right takes too long to compute.

from sympy import *
Vmax, Amax = symbols('Vmax, Amax')
eq1 = Vmax*5.041667 + (Vmax**2/Amax)*sympy.exp((-Amax/Vmax)*5.041667) - (Vmax**2/Amax)
eq2 = Vmax*2.897628 + (Vmax**2/Amax)*sympy.exp((-Amax/Vmax)*2.897628) - (Vmax**2/Amax)
solve([eq1-40, eq2-20], (Vmax, Amax))

Any idea what I can do to fix my code so I can get the right answer?

Answer 1

You've misunderstood how nsolve works. The third argument is an initial guess that is used to start the root-finding algorithm. Here [1, 1] will work fine as the guess:

In [26]: from sympy import symbols, Eq, nsolve, exp                                                                                                            

In [27]: Vmax, Amax = symbols('Vmax, Amax')                                                                                       

In [28]: eq1 = Eq(Vmax*5.041667 + (Vmax**2/Amax)*exp((-Amax/Vmax)*5.041667) - (Vmax**2/Amax), 40) 
    ...: eq2 = Eq(Vmax*2.897628 + (Vmax**2/Amax)*exp((-Amax/Vmax)*2.897628) - (Vmax**2/Amax), 20)                           

In [29]: nsolve([eq1, eq2], [Amax, Vmax], [1, 1])                                                                                 
Out[29]: 
⎡11.8641453843429⎤
⎢                ⎥
⎣9.41244210257784⎦

Note that this only finds one solution starting from any particular initial guess. It is possible that that a system like this has more than one solution.