I am given a differential equation in $x \in [0, L]$
\begin{align} \lambda\frac{d^2u}{dx^2} + q = 0 \end{align}
where $q = a + bx$. The solution is subject to the boundary conditions
\begin{align} u(0) = 0 \ \ and \ \ \lambda\frac{du}{dx}\bigg |_{x = L} = 0 \end{align}
Now I want to find an approximate solution using a three-node element. The shape function is given as
\begin{align} \mathbf{N} = \begin{bmatrix} -\frac{1}{2}s(1-s) & (1-s^2) & \frac{1}{2}s(1+s) \end{bmatrix} \end{align}
The task is the solve the system but what I am wondering is what I need to calculate in order to do this? For example I have seen equations such as
\begin{align} \mathbf{k} = \int_{-1}^1\mathbf{B^TBJ}ds \end{align}
and
\begin{align} \mathbf{m} = \int_{-1}^1\mathbf{N^TNJ}ds \end{align}
But how should I use this information in order to achieve the final solution?
