second order of accuracy approximation formula

Assuming y (x) is a smooth function defined on the interval [0; 1] ; obtain a second order of accuracy approximation formula for y''' (1): You can make use of MATLAB software to find the unknown coeficients. Write down the formula as a result.

clear
clc
syms x y a b c d e A B C D h
y=@(x,h) a+b*h+c*h^2/2+d*h^3/6;                             % general form of Taylor series where a=y(0),b=y'(0)..etc are non-zero constants
t=A*y(x,0)+B*y(x,h)+C*y(x,2*h)+D*y(x,3*h);                  % Find the linear sum of Ay(x)+By(x+h)..etc
eqn1=simplify(subs(t,h,0)/a);                               % dividing by constant to get the equation
eqn2=simplify((subs(diff(t,h,1),h,0))/b);
eqn3=simplify((subs(diff(t,h,2),h,0))/c);
eqn4=simplify((subs(diff(t,h,3),h,0))/d);
s=solve([eqn1==0,eqn2==0,eqn3==1/h^2,eqn4==1],[A,B,C,D]);   % solve the system of equations for A,B,C,D
%solve the system so that the coefficients of y,y',y'' are all 0
%except y''' whose coeff is 1
sol=[s.A;s.B;s.C;s.D]

I made a solution like this, is it correct?

[Torsten]

No.

Compare your coefficients with the correct values tabulated here:

https://en.wikipedia.org/wiki/Finite_difference_coefficient