数值解决python或Matlab中的超越方程

Question

I have a Euler-Bernoulli equation I'm trying to solve where:

q(x) = a*x + b + w(x)

Euler-Bernoulli equation:

E * I * diff(w(x), x, x, x, x) = q(x) 

I don't really know if this is Transcendental, but I have E, I, a, b and know my integration limits too (0 to H). I have numbers for E, I, a, b and H.

How can I obtain points of w(x),x varying x from 0 to H?

Answer 1

If you wanted to solve it symbolically, you could, with eg sympy :

from __future__ import division
from sympy import *
x = symbols('x')
w = symbols('w', cls=Function)
a,b,E,J =  symbols('a b E J')
equ = E*J*diff(w(x),x,4) - a*x -b - w(x)
dsolve(equ, w(x))
# This generates a function that is too generic and too big to copy-paste
# Let's make some assumptions
J = Symbol('J', real=True, positive=True)
E = Symbol('E', real=True, positive=True)
equ = E*J*diff(w(x),x,4) - a*x -b - w(x)
dsolve(equ, w(x))

which results in:

               -x                 x                                                           
           ───────────       ───────────                                                      
           4 ___ 4 ___       4 ___ 4 ___                                                      
           ╲╱ E ⋅╲╱ J        ╲╱ E ⋅╲╱ J          ⎛     x     ⎞         ⎛     x     ⎞          
w(x) = C₁⋅ℯ            + C₂⋅ℯ            + C₃⋅sin⎜───────────⎟ + C₄⋅cos⎜───────────⎟ - a⋅x - b
                                                 ⎜4 ___ 4 ___⎟         ⎜4 ___ 4 ___⎟          
                                                 ⎝╲╱ E ⋅╲╱ J ⎠         ⎝╲╱ E ⋅╲╱ J ⎠          

Which you could probably even simplify further, given extra information about your boundary conditions. You'll need them in any case, because you still have 4 unknown coefficients.

Answer 2

Let me add a solution using the Symbolic toolbox in Matlab:

syms q x w(x) a b E I
q = a*x+b+w(x);
sol = dsolve(E*I*diff(diff(diff(diff(w)))) == q)
sol_part = subs(sol,[E I],[2e5 100]) % etc.

You can add the boundary conditions as separate equations to the dsolve command, thus forming a system of equations.