Finite element method for the construction of the approximation of the solution

[mathmari]

Hey!

Given the following two-point problem:
$$-y''(x)+(by)'(x)=f(x), \forall x \in [0,1]$$
$$y(0)=0, y'(1)=my(1)$$
where $ b \in C^1([0,1];R), f \in C([0,1];R)$ and $ m \in R$ a constant.
Give a finite element method for the construction of the approximation of the solution $y$ of the problem above, where the finite element space ($S$) consists of continuous and partially linear functions.

My idea is the following:
$ u \in S:$
$$ -\int_0^1{u''g}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$$ -u'g|_0^1+ \int_0^1{u'g'}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$$-mu(1)+g(1)+ \int_0^1{u'g'}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$ \forall g \in S$

Could you tell me if this is correct?

 

[mathmari]

To find the method we take a function $g$ of $S$, right? Does this function satisfy the conditions of the problem? I mean $g(0)=0, g'(1)=mg(1)$... Or is there an other way to find the method?