Method of Undetermined Coefficients I am trying to solve a problem using method of undetermined coefficients to derive a second order scheme for ux using three points, c1, c2, c3 in the following way:
ux = c1*u(x) + c2*u(x - h) + c3*u(x - 2h)
Now second order scheme just means to solve the equation for the second order derivative, am I right?
I understand how this problem works for actual numerical functions, but I am unsure how to go about it when everything is theoretical and just variables.
Thanks for some help

Having a second order scheme means that it's accurate for polynomials up to and including second degree. The scheme should calculate the first order derivative $u_x$, as the formula says. 
It suffices to make sure that the scheme is accurate for $1$, $x$, and $x^2$; then it will  work for all second-degree polynomials by linearity. 
To make it work for the function $u(x)=1$, we need
$$ 0= c_1+c_2+c_3
\tag1$$
To make it  work for the linear function $u(x)=x$, we   need
$$ 1 = c_1 x +c_2(x-h) +c_3(x-2h)
\tag2$$
which in view of (1) simplifies to 
$$ 1 = c_2(-h) +c_3(-2h)
\tag{2'}$$
And to make it work for the quadratic function $u(x)=x^2$, we need 
$$ 2x = c_1 x^2 +c_2(x-h)^2 +c_3(x-2h)^2
\tag3$$
which in view of (1) and (2') simplifies to 
$$ 0 =  c_2h^2 +c_3(4h^2)
\tag{3'}$$
Now you can solve the linear system (1), (2') and (3') for the unknowns $c_1,c_2,c_3$.

This may not be the quickest solution, but it's the most concrete one that I could think of. 
