How do I solve a non-linear equation in Sympy?

Question

How do I solve a non-linear equation in SymPy which is of the form

y = P*x + Q + sqrt(S*x + T)

where I know y(0), y'(0), y(c), y'(c). I want to find P, Q, S and T. and represent y as a function of x.

I am getting very confused with the documentation. Please help.

Answer 1

NOTE: My sympy hanged on your original equation of y = P*x + Q + sqrt(S*x + T). I will be using y = P*x + Q + x*x*(S*x + T) just to be able to demonstrate how the sympy solver works (when it works).

Strategy:

• Express y as a function of the other variables (x, P, Q, S, T)
• Differentiate y
• Set up 4 equations using the known constants ( 0, c, y(0), y(c), y'(0), y'(c) )
• Use sympy solve
• Print each possible solution (if there are any)

Code:

# Set up variables and equations
x, y, P, Q, S, T,  = sympy.symbols('x y P Q S T')
c, y_0, y_c, dy_0, dy_c = sympy.symbols('c y_0 y_c dy_0 dy_c')
eq_y = P * x + Q + x * x * (S * x + T)
eq_dy = eq_y.diff(x)

# Set up simultaneous equations that sympy will solve
equations = [
    (y_0 - eq_y).subs(x, 0),
    (dy_0 - eq_dy).subs(x, 0),
    (y_c - eq_y).subs(x, c),
    (dy_c - eq_dy).subs(x, c)
]

# Solve it for P, Q, S and T
solution_set = sympy.solve(equations, P, Q, S, T, set = True) 

# Extract names, individual solutions and print everything
names = solution_set[0]
solutions = list(solution_set[1])
for k in range(len(solutions)):
    print('Solution #%d' % (k+1))
    for k2, name in enumerate(names):
        print('\t%s: %s' % (name, solutions[k][k2]) )

Output:

Solution #1
    P: dy_0
    Q: y_0
    S: (c*(dy_0 + dy_c) + 2*y_0 - 2*y_c)/c**3
    T: (-c*(2*dy_0 + dy_c) - 3*y_0 + 3*y_c)/c**2

You can now use one of these solutions and do another .subs(...) to get y as a function purely consisting of your constants and x.

As for your original equation... I wonder if someone should file a bug report for sympy so they can improve upon it... :)