Given the problem
\begin{align} &u''− x^2u(u − 1) = 0,\\ &u(1) = 2,\\ &u(3) = 4, \end{align}
solve it with a finite difference method with the interval divided into $N+1$ equal intervals between $1 \leq x \leq 3$.
- First in detail write the equations when $N=3$.
- Draw the interval and mark the discretization points for the differential equation in the $N$ inner points for arbitrary $N$-value.
- Write the equations $f(u) = 0$. Write a MATLAB function that for a given $u$ calculates $f(u)$.
- Use global values for the boundary values.
What I can do is transform the differential equation to first order:
$ v_1=u,\\ v_2=u',\\ v_2'=u''. $
Hence
$ v_2'-x^2v_1(v_1-1)=0\\ v_1(1)=2\\ v_1(3)=4 $
But I don't know how to continue.