How to solve for 1D non homogenous ODE by Finite element method

I am unable to input the dirichlet condition into this non homogenous ode. Please point out my mistake.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%       -u" + u = - 5x + 4   0 < x <= 1       %%%%
%%%%            u(0) = 0, u(1) = 2                         %%%%
%%%%     Exact solution u = (exp(x)*(3*exp(1) + 4))/(exp(2) - 1) - 5*x - (exp(-x)*(3*exp(1) + 4*exp(2)))/(exp(2) - 1) + 4                   %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
format long;
N = 20;
x0 = 0;
xN = 1;
h = 1/N;
for j = 1:N+1
    X(j) = x0 + (j-1)*h;
end
f = @(x)(- 5*x +4);
A = zeros(N+1,N+1);
F = zeros(N+1,1);
%%%% Local stiffness matrix %%%%
a = [1/h -1/h ; -1/h 1/h] ; 
for i=1:N
    phi1 = @(x)(X(i+1)-x)/h;            %%%% linear basis function %%%%
    phi2 = @(x)(x-X(i))/h;              %%%% linear basis function %%%%
    f1 = @(x)f(x)*phi1(x);              %%%% integrand for load vector %%%%
    f2 = @(x)f(x)*phi2(x);              %%%% integrand for load vector %%%%
    v(1,i) = gauss(f1,X(i),X(i+1),1);   %%%% element wise values of
    v(2,i) = gauss(f2,X(i),X(i+1),1);   %%%% load vector
end

Unrecognized function or variable 'gauss'.

%%%% Assembling %%%%
for i=1:N
    A([i i+1],[i i+1]) = A([i i+1],[i i+1]) + a;
    F([i i+1],1) = F([i i+1],1) + v([1 2],i);
end
%%%% Dirichlet Boundary condition %%%%
% F(N+1,1) = F(N+1,1)+2;
fullnodes = [1:N+1];
%%%%% Dirichlet boundary condition %%%%%
freenodes=setdiff(fullnodes,[1]);
Uh = zeros(N+1,1);
Uh(N+1,1) = 2;
%%%% Approximate solution %%%%
Uh(freenodes)=A(freenodes,freenodes)\F(freenodes,1);
%%%% Exact solution %%%%
U = zeros(N+1,1);
for i =1:N+1
    U(i) = (exp(X(i))*(3*exp(1) + 4))/(exp(2) - 1) - 5*X(i) - (exp(-X(i))*(3*exp(1) + 4*exp(2)))/(exp(2) - 1) + 4;
end
error = max(abs(U-Uh));
[U Uh]
plot(X,U,X,Uh,'o')

Answer 1

[Torsten]

I usually set

A(1,1) = 1, F(1) = 0, A(N+1,N+1) = 1, F(N+1) = 2

and only loop from 2 to N here:

%%%% Assembling %%%%
for i=1:N
    A([i i+1],[i i+1]) = A([i i+1],[i i+1]) + a;
    F([i i+1],1) = F([i i+1],1) + v([1 2],i);
end

I don't know if this is "Finite-Element-like".